#### Sample records for binomial distributions

1. Learning Poisson Binomial Distributions

Daskalakis, Constantinos; Diakonikolas, Ilias; Servedio, Rocco A

2015-01-01

We consider a basic problem in unsupervised learning: learning an unknown Poisson binomial distribution. A Poisson binomial distribution (PBD) over TeX is the distribution of a sum of TeX independent Bernoulli random variables which may have arbitrary, potentially non-equal, expectations. These distributions were first studied by Poisson (Recherches sur la Probabilitè des jugements en matié criminelle et en matiére civile. Bachelier, Paris, 1837) and are a natural TeX -parameter generalizatio...

2. Learning Poisson Binomial Distributions

2011-01-01

We consider a basic problem in unsupervised learning: learning an unknown \\emph{Poisson Binomial Distribution} over $\\{0,1,...,n\\}$. A Poisson Binomial Distribution (PBD) is a sum $X = X_1 + ... + X_n$ of $n$ independent Bernoulli random variables which may have arbitrary expectations. We work in a framework where the learner is given access to independent draws from the distribution and must (with high probability) output a hypothesis distribution which has total variation distance at most $\\eps$ from the unknown target PBD. As our main result we give a highly efficient algorithm which learns to $\\eps$-accuracy using $\\tilde{O}(1/\\eps^3)$ samples independent of $n$. The running time of the algorithm is \\emph{quasilinear} in the size of its input data, i.e. $\\tilde{O}(\\log(n)/\\eps^3)$ bit-operations (observe that each draw from the distribution is a $\\log(n)$-bit string). This is nearly optimal since any algorithm must use $\\Omega(1/\\eps^2)$ samples. We also give positive and negative results for some extensi...

3. A new Markov Binomial distribution.

Omey, Edward; Minkova, Leda D.

2011-01-01

In this paper, we introduce a two state homogeneous Markov chain and define a geometric distribution related to this Markov chain. We define also the negative binomial distribution similar to the classical case and call it NB related to interrupted Markov chain. The new binomial distribution is related to the interrupted Markov chain. Some characterization properties of the Geometric distributions are given. Recursion formulas and probability mass functions for the NB distribution and the new...

4. Distinguishing between Binomial, Hypergeometric and Negative Binomial Distributions

Wroughton, Jacqueline; Cole, Tarah

2013-01-01

Recognizing the differences between three discrete distributions (Binomial, Hypergeometric and Negative Binomial) can be challenging for students. We present an activity designed to help students differentiate among these distributions. In addition, we present assessment results in the form of pre- and post-tests that were designed to assess the…

5. A Markov-binomial distribution

Santos, J.; S. Van Gulck; OMEY, E.

2007-01-01

Let ${X_{i},igeq 1}$ denote a sequence of $left{ 0,1 ight}$%-variables and suppose that the sequence forms a {sc Markov} Chain. In the paperwe study the number of successes $S_{n}=X_{1}+X_{2}+cdots+X_{n}$ and we studythe number of experiments $Y(r)$ up to the $r$-$th$ success. In the i.i.d.case $S_{n}$ has a binomial distribution and $Y(r)$ has a negative binomialdistribution and the asymptotic behaviour is well known. In the more general{sc Markov} chain case, we prove a central limit theor...

6. A Characterization of the Negative Binomial Distribution

Kolev, Nikolay(Department of Physics, University of Regina, SK S4S 0A2, Canada); Minkova, Leda

2000-01-01

Only a few characterizations have been obtained in literatute for the negative binomial distribution (see Johnson et al., Chap. 5, 1992). In this article a characterization of the negative binomial distribution related to random sums is obtained which is motivated by the geometric distribution characterization given by Khalil et al. (1991). An interpretation in terms of an unreliable system is given.

7. The Negative Binomial Distribution in Quantum Physics

Söderholm, Jonas; Inoue, Shuichiro

2009-06-01

We give examples of situations where the negative binomial distribution has appeared in quantum physics since its debut in the work of Planck. Several of its properties are reviewed, and Mandel's Q-parameter is shown to play an interesting role. The photon-pair distributions of squeezed vacuum and squeezed single-photon states are identified as negative binomial.

8. On approximation of Markov binomial distributions

Xia, Aihua; 10.3150/09-BEJ194

2010-01-01

For a Markov chain $\\mathbf{X}=\\{X_i,i=1,2,...,n\\}$ with the state space $\\{0,1\\}$, the random variable $S:=\\sum_{i=1}^nX_i$ is said to follow a Markov binomial distribution. The exact distribution of $S$, denoted $\\mathcal{L}S$, is very computationally intensive for large $n$ (see Gabriel [Biometrika 46 (1959) 454--460] and Bhat and Lal [Adv. in Appl. Probab. 20 (1988) 677--680]) and this paper concerns suitable approximate distributions for $\\mathcal{L}S$ when $\\mathbf{X}$ is stationary. We conclude that the negative binomial and binomial distributions are appropriate approximations for $\\mathcal{L}S$ when $\\operatorname {Var}S$ is greater than and less than $\\mathbb{E}S$, respectively. Also, due to the unique structure of the distribution, we are able to derive explicit error estimates for these approximations.

9. On hypergeometric generalized negative binomial distribution

Ghitany, M. E.; Al-Awadhi, S. A.; S. L. Kalla

2002-01-01

It is shown that the hypergeometric generalized negative binomial distribution has moments of all positive orders, is overdispersed, skewed to the right, and leptokurtic. Also, a three-term recurrence relation for computing probabilities from the considered distribution is given. Application of the distribution to entomological field data is given and its goodness-of-fit is demonstrated.

10. On hypergeometric generalized negative binomial distribution

M. E. Ghitany

2002-01-01

Full Text Available It is shown that the hypergeometric generalized negative binomial distribution has moments of all positive orders, is overdispersed, skewed to the right, and leptokurtic. Also, a three-term recurrence relation for computing probabilities from the considered distribution is given. Application of the distribution to entomological field data is given and its goodness-of-fit is demonstrated.

11. Methods of Estimation of Generalized Negative Binomial Distribution

Kumar, Abhay; Bharti, Ramesh Chandra; Singh, Sree Kant; Mishra, Amarendra; Singh, Krishna Murari

2014-01-01

The negative binomial distribution was perhaps the first probability distribution, considered in statistics, whose variance is larger than its mean. On account of wide variety of available discrete distributions, the research workers in applied fields have begun to wonder which distribution would be most suitable one in a particular case and how to choose it. Generalized Negative Binomial Distribution (GNBD) reduces the binomial or the negative binomial distribution as particular cases and co...

12. Library Book Circulation and the Beta-Binomial Distribution.

Gelman, E.; Sichel, H. S.

1987-01-01

Argues that library book circulation is a binomial rather than a Poisson process, and that individual book popularities are continuous beta distributions. Three examples demonstrate the superiority of beta over negative binomial distribution, and it is suggested that a bivariate-binomial process would be helpful in predicting future book…

13. On approximation of Markov binomial distributions

Xia, Aihua; Zhang, Mei

2009-01-01

For a Markov chain $\\mathbf{X}=\\{X_i,i=1,2,...,n\\}$ with the state space $\\{0,1\\}$, the random variable $S:=\\sum_{i=1}^nX_i$ is said to follow a Markov binomial distribution. The exact distribution of $S$, denoted $\\mathcal{L}S$, is very computationally intensive for large $n$ (see Gabriel [Biometrika 46 (1959) 454--460] and Bhat and Lal [Adv. in Appl. Probab. 20 (1988) 677--680]) and this paper concerns suitable approximate distributions for $\\mathcal{L}S$ when $\\mathbf{X}$ is stationary. We...

14. Generating functions for generalized binomial distributions

Bergeron, H.; Curado, E. M. F.; Gazeau, J. P.; Rodrigues, Ligia M. C. S.

2012-10-01

In a recent article generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal probability distributions was a key point to allow to give them a statistical interpretation in terms of probabilities. In this article we present an approach based on generating functions that solves the previous difficulties. Our main theorem makes explicit the conditions under which those formal probability distributions are always non-negative. Therefore, the constraints of non-negativeness are automatically fulfilled giving a complete characterization in terms of generating functions. A large number of analytical examples becomes available.

15. Does the negative binomial distribution add up?

Grafen, A; Woolhouse, M E

1993-12-01

The negative binomial distribution (NBD) is widely used to describe the distribution of parasitic helminths in a number of host individuals and has proved a useful, though possibly overused, empirical and theoretical device. It is therefore important that the limits to the applicability of the NBD be clearly defined. In this paper, Alan Grafen and Mark Woolhouse consider applications of the NBD in situations where either the host or parasite population can be divided into subpopulations of different types (eg. by age, sex or genotype), and they describe the relationships between the frequency distributions relevant to the different subpopulations and those relevant to the total population. PMID:15463698

16. Negative Binomial and Multinomial States: probability distributions and coherent states

Fu, Hong-Chen; Sasaki, Ryu

1996-01-01

Following the relationship between probability distribution and coherent states, for example the well known Poisson distribution and the ordinary coherent states and relatively less known one of the binomial distribution and the $su(2)$ coherent states, we propose interpretation'' of $su(1,1)$ and $su(r,1)$ coherent states in terms of probability theory''. They will be called the negative binomial'' (multinomial'') states'' which correspond to the negative'' binomial (multinomial)...

17. Negative Binomial-Lindley Distribution and Its Application

Hossein Zamani

2010-01-01

Full Text Available Problem statement: The modeling of claims count is one of the most important topics in actuarial theory and practice. Many attempts were implemented in expanding the classes of mixed and compound distributions, especially in the distribution of exponential family, resulting in a better fit on count data. In some cases, it is proven that mixed distributions, in particular mixed Poisson and mixed negative binomial, provided better fit compared to other distributions. Approach: In this study, we introduce a new mixed negative binomial distribution by mixing the distributions of negative binomial (r,p and Lindley (θ, where the reparameterization of p = exp(-λ is considered. Results: The closed form and the factorial moment of the new distribution, i.e., the negative binomial-Lindley distribution, are derived. In addition, the parameters estimation for negative binomial-Lindley via the method of moments (MME and the Maximum Likelihood Estimation (MLE are provided. Conclusion: The application of negative binomial-Lindley distribution is carried out on two samples of insurance data. Based on the results, it is shown that the negative binomial-Lindley provides a better fit compared to the Poisson and the negative binomial for count data where the probability at zero has a large value.

18. Wigner Function of Density Operator for Negative Binomial Distribution

HE Min-Hua; XU Xing-Lei; ZHANG Duan-Ming; LI Hong-Qi; PAN Gui-Jun; YIN Yan-Ping; CHEN Zhi-Yuan

2008-01-01

By using the technique of integration within an ordered product (IWOP) of operator we derive Wigner function of density operator for negative binomial distribution of radiation field in the mixed state case, then we derive the Wigner function of squeezed number state, which yields negative binomial distribution by virtue of the entangled state representation and the entangled Wigner operator.

19. Convergence properties of the q-deformed binomial distribution

Martin Zeiner

2010-03-01

Full Text Available We consider the $q$-deformed binomial distribution introduced by{sc S. C. Jing:} {it The {$q$}-deformed binomial distribution and its asymptotic behaviour,}J. Phys. A {f 27} (2 (1994, 493--499and{sc W. S. Chung} et al: {it {$q$}-deformed probability and binomial distribution,} Internat. J. Theoret. Phys.{f 34} (11 (1995, 2165--2170and establish several convergence results involvingthe Euler and the exponential distribution; some of them are $q$-analogues of classical results.

20. The negative binomial distribution in quark jets with fixed flavour

Alberto GiovanniniTurin U. & INFN, Turin; Sergio Lupia(Munich, Max Planck Inst.); Roberto Ugoccioni(Lund U.)

2015-01-01

We show that both the multiplicity distribution and the ratio of factorial cumulants over factorial moments for 2-jet events in e+e- annihilation at the Z^0 peak can be well reproduced by the weighted superposition of two negative binomial distributions, associated to the contribution of $b\\bar b$ and light flavoured events respectively. The negative binomial distribution is then suggested to describe the multiplicity distribution of 2-jet events with fixed flavour.

1. Negative Binomial Distribution and the multiplicity moments at the LHC

Praszalowicz, Michal

2011-10-01

In this work we show that the latest LHC data on multiplicity moments C2-C5 are well described by a two-step model in the form of a convolution of the Poisson distribution with energy-dependent source function. For the source function we take Γ Negative Binomial Distribution. No unexpected behavior of Negative Binomial Distribution parameter k is found. We give also predictions for the higher energies of 10 and 14 TeV.

2. Negative Binomial Distribution and the multiplicity moments at the LHC

Praszalowicz, Michal

2011-01-01

In this work we show that the latest LHC data on multiplicity moments $C_2-C_5$ are well described by a two-step model in the form of a convolution of the Poisson distribution with energy-dependent source function. For the source function we take $\\Gamma$ Negative Binomial Distribution. No unexpected behavior of Negative Binomial Distribution parameter $k$ is found. We give also predictions for the higher energies of 10 and 14 TeV.

3. Self-Similarity of the Negative Binomial Multiplicity Distributions

Calucci, Giorgio; Treleani, Daniele

1997-01-01

The negative binomial distribution is self similar: If the spectrum over the whole rapidity range gives rise to a negative binomial, in absence of correlation and if the source is unique, also a partial range in rapidity gives rise to the same distribution. The property is not seen in experimental data, which are rather consistent with the presence of a number of independent sources. When multiplicities are very large self similarity might be used to isolate individual sources is a complex pr...

4. Estimation of Log-Linear-Binomial Distribution with Applications

Elsayed Ali Habib

2010-01-01

Full Text Available Log-linear-binomial distribution was introduced for describing the behavior of the sum of dependent Bernoulli random variables. The distribution is a generalization of binomial distribution that allows construction of a broad class of distributions. In this paper, we consider the problem of estimating the two parameters of log-linearbinomial distribution by moment and maximum likelihood methods. The distribution is used to fit genetic data and to obtain the sampling distribution of the sign test under dependence among trials.

5. On intervened negative binomial distribution and some of its properties

C. Satheesh Kumar

2013-05-01

Full Text Available Here we develop a new class of discrete distribution namely intervened negative binomial distribution and derive its probability generating function, mean, variance and an expression for its factorial moments. Estimation of the parameters of the distribution is described and the distribution has been fitted to a well known data set.

6. On intervened negative binomial distribution and some of its properties

Satheesh Kumar, C.; S. Sreejakumari

2013-01-01

Here we develop a new class of discrete distribution namely intervened negative binomial distribution and derive its probability generating function, mean, variance and an expression for its factorial moments. Estimation of the parameters of the distribution is described and the distribution has been fitted to a well known data set.

7. Hadronic multiplicity distributions: the negative binomial and its alternatives

We review properties of the negative binomial distribution, along with its many possible statistical or dynamical origins. Considering the relation of the multiplicity distribution to the density matrix for boson systems, we re-introduce the partially coherent laser distribution, which allows for coherent as well as incoherent hadronic emission from the k fundamental cells, and provides equally good phenomenological fits to existing data. The broadening of non-single diffractive hadron-hadron distributions can be equally well due to the decrease of coherence with increasing energy as to the large (and rapidly decreasing) values of k deduced from negative binomial fits. Similarly the narrowness of e+-e- multiplicity distribution is due to nearly coherent (therefore nearly Poissonian) emission from a small number of jets, in contrast to the negative binomial with enormous values of k. 31 refs

8. A continuous version of the negative binomial distribution

Nimai Kumar Chandra

2013-05-01

Full Text Available While discretization of continuous distributions have been attempted for many life distributions the reverse has hardly been attempted. The present endeavor is to establish a reverse relationship by offering a continuous counter part of a discrete distribution namely negative binomial distribution. Different properties of this distribution have been established for a special choice of the parametric value covering class properties, ordering and mean residual life.

9. A continuous version of the negative binomial distribution

Nimai Kumar Chandra; Dilip Roy

2013-01-01

While discretization of continuous distributions have been attempted for many life distributions the reverse has hardly been attempted. The present endeavor is to establish a reverse relationship by offering a continuous counter part of a discrete distribution namely negative binomial distribution. Different properties of this distribution have been established for a special choice of the parametric value covering class properties, ordering and mean residual life.

10. Negative Binomial-Lindley Distribution and Its Application

Hossein Zamani; Noriszura Ismail

2010-01-01

Problem statement: The modeling of claims count is one of the most important topics in actuarial theory and practice. Many attempts were implemented in expanding the classes of mixed and compound distributions, especially in the distribution of exponential family, resulting in a better fit on count data. In some cases, it is proven that mixed distributions, in particular mixed Poisson and mixed negative binomial, provided better fit compared to other distributions. Approach: In this study, we...

11. Correlation Structures of Correlated Binomial Models and Implied Default Distribution

Mori, S.; K. Kitsukawa; M. Hisakado

2006-01-01

We show how to analyze and interpret the correlation structures, the conditional expectation values and correlation coefficients of exchangeable Bernoulli random variables. We study implied default distributions for the iTraxx-CJ tranches and some popular probabilistic models, including the Gaussian copula model, Beta binomial distribution model and long-range Ising model. We interpret the differences in their profiles in terms of the correlation structures. The implied default distribution h...

12. Multiplicity distributions in e+e- annihilation into hadrons and the extended modified negative binomial

Tchikilev, O. G.

1997-01-01

It is shown that simple extension of the modified negative binomial distribution describes negatively charged particle multiplicity distributions in e+e- annihilation, measured in the whole phase space, as well as the modified negative binomial.

13. Statistical Inference for a Class of Multivariate Negative Binomial Distributions

Rubak, Ege H.; Møller, Jesper; McCullagh, Peter

This paper considers statistical inference procedures for a class of models for positively correlated count variables called -permanental random fields, and which can be viewed as a family of multivariate negative binomial distributions. Their appealing probabilistic properties have earlier been...... studied in the literature, while this is the first statistical paper on -permanental random fields. The focus is on maximum likelihood estimation, maximum quasi-likelihood estimation and on maximum composite likelihood estimation based on uni- and bivariate distributions. Furthermore, new results for...

14. Negative Binomial Distribution and the multiplicity moments at the LHC

Praszalowicz, Michal

2011-01-01

In this work we show that the latest LHC data on multiplicity moments $C_2-C_5$ are well described by a two-step model in the form of a convolution of the Poisson distribution with energy-dependent source function. For the source function we take $\\Gamma$ Negative Binomial Distribution. We give also predictions for the higher energies of 10 and 14 TeV.

15. Mixed Negative Binomial Distribution by Weighted Gamma Mixing Distribution

Stoynov, Pavel

2011-01-01

Павел Т. Стойнов - В тази работа се разглежда отрицателно биномното разпределение, известно още като разпределение на Пойа. Предполагаме, че смесващото разпределение е претеглено гама разпределение. Изведени са вероятностите в някои частни случаи. Дадени са рекурентните формули на Панжер. In this paper the mixed negative binomial distribution, known also as P´olya distribution is considered. We suppose that the mixing distribution is a weighted Gamma distribution. We derive the probabil...

16. KNO scaling function of modified negative binomial distribution

Nakajima, N; Biyajima, M.; Suzuki, N.

1996-01-01

We investigate the KNO scaling function of the modified negative binomial distribution (MNBD), because this MNBD can explain the oscillating behaviors of the cumulant moment observed in $e^+e^-$ annihilations and in hadronic collisions. By using a straightforward method and the Poisson transform we derive the KNO scaling function from the MNBD. The KNO form of experimental data in $e^{+}e^{-}$ collisions and hadronic collisions are analyzed by the KNO scaling function of the MNBD and that of ...

17. KNO scaling function of modified negative binomial distribution

Nakajima, Noriaki; Biyajima, Minoru; Suzuki, Naomichi

1997-01-01

We investigate the KNO scaling function of the modified negative binomial distribution (MNBD), because this MNBD can explain the oscillating behaviors of the cumulant moment observed in ez e{ annihilations and in hadronic collisions. By using a straightforward method and the Poisson transform we derive the KNO scaling function from the MNBD. The KNO form of experimental data in ez e{ collisions and hadronic collisions are analyzed by the KNO scaling function of the MNBD and that of the nega...

18. Comparison of multiplicity distributions to the negative binomial distribution in muon-proton scattering

The multiplicity distributions of charged hadrons produced in the deep inelastic muon-proton scattering at 280 GeV are analysed in various rapidity intervals, as a function of the total hadronic centre of mass energy W ranging from 4-20 GeV. Multiplicity distributions for the backward and forward hemispheres are also analysed separately. The data can be well parameterized by binomial distributions, extending their range of applicability to the case of lepton-proton scattering. The energy and the rapidity dependence of the parameters is presented and a smooth transition from the binomial distribution via Poissonian to the ordinary binomial is observed. (orig.)

19. Statistical inference for a class of multivariate negative binomial distributions

Rubak, Ege Holger; Møller, Jesper; McCullagh, Peter

This paper considers statistical inference procedures for a class of models for positively correlated count variables called α-permanental random fields, and which can be viewed as a family of multivariate negative binomial distributions. Their appealing probabilistic properties have earlier been...... studied in the literature, while this is the first statistical paper on α-permanental randomfields. The focus is on maximum likelihood estimation, maximum quasi-likelihood estimation and on maximum composite likelihood estimation based on uni- and bivariate distributions. Furthermore, new results for α...

20. The Parameterized Complexity Analysis of Partition Sort for Negative Binomial Distribution Inputs

Singh, Niraj Kumar; Chakraborty, Soubhik

2012-01-01

The present paper makes a study on Partition sort algorithm for negative binomial inputs. Comparing the results with those for binomial inputs in our previous work, we find that this algorithm is sensitive to parameters of both distributions. But the main effects as well as the interaction effects involving these parameters and the input size are more significant for negative binomial case.

1. Multiplicities in Ultrarelativistic - Collisions and Negative Binomial Distribution FITS

Prorok, Dariusz

Likelihood ratio tests are performed for the hypothesis that charged-particle multiplicities measured in proton-(anti)proton collisions at √ {s} = 0.9 and 2.36 TeV are distributed according to the negative binomial form. Results indicate that the hypothesis should be rejected in the all cases of ALICE-LHC measurements in the limited pseudorapidity windows, whereas should be accepted in the corresponding cases of UA5 data. Possible explanations of that and of the disagreement with the least-squares fitting method are given.

2. On Size-Biased Negative Binomial Distribution and its Use in Zero-Truncated Cases

2009-01-01

A size-biased negative binomial distribution, a particular case of the weighted negative binomial distribution, taking the weights as the variate values has been defined. A Bayes' estimator of size-biased negative binomial distribution (SBNBD) has been obtained by using non-informative and gamma prior distributions. Also comparison has been made of this estimator with the corresponding maximum likelihood estimator (MLE) with the help of R- Software.

3. Moody's Correlated Binomial Default Distributions for Inhomogeneous Portfolios

Mori, S; Kitsukawa, K

2006-01-01

This paper generalizes Moody's correlated binomial default distribution for homogeneous credit portfolio, which is introduced by Witt, to the case of inhomogeneous portfolios. As inhomogeneous portfolios, we consider two cases. In the first case, we treat a portfolio whose assets have uniform default correlation and non-uniform default probabilities. We obtain the default probability distribution and study the effect of the inhomogeneity on it. The second case corresponds to a portfolio with inhomogeneous default correlation. Assets are categorized in several different sectors and the inter-sector and intra-sector correlations are not the same. We construct the joint default probabilities and obtain the default probability distribution. We show that as the number of assets in each sector decreases, inter-sector correlation becomes more important than the intra-sector one. We study the maximum values of the inter-sector default correlation. In the modeling of portfolio credit risk, it is possible to incorporat...

4. The zero inflated negative binomial – Crack distribution: some properties and parameter estimation

Pornpop Saengthong; Winai Bodhisuwan; Ampai Thongteeraparp

2015-01-01

The zero inflated negative binomial-Crack (ZINB-CR) distribution is a mixture of Bernoulli distribution and negative binomial-Crack (NB-CR) distribution, which is an alternative distribution for the excessive zero counts and overdispersion. In this paper, some properties of the ZINB-CR distribution are discussed. Statistical inference of the parameters is derived by maximum likelihood estimation (MLE) and the method of moments (MM). Monte Carlo Simulations are used to evaluate the ...

5. Binomial Distribution Sample Confidence Intervals Estimation 5. Odds Ratio

Sorana BOLBOACĂ

2004-02-01

Full Text Available Evaluation of the strength of association between predisposing or causal factors and disease can be express as odds ratio in case-control studies. In order to interpret correctly a point estimation of odds ratio we need to look also to its confidence intervals quality. The aim of this paper is to introduce three new methods of computing the confidence intervals, R2AC, R2Binomial, and R2BinomialC, and compare the performances with the asymptotic method called R2Wald.In order to assess the methods a PHP program was develop. First, the upper and lower confidence boundaries for all implemented methods were computes and graphically represented. Second, the experimental errors, standard deviations of the experimental errors and deviation relative to the imposed significance level α = 5% were assessed. Estimating the experimental errors and standard deviations at central point for given sample sizes was the third criterion. The R2Wald and R2AC methods were assessed using random binomial variables (X, Y and sample sizes (m, n from 4 to 1000. The methods based on the original method Binomial adjusted for odds ratio (R2Binomial, R2BinomialC functions obtain systematically the lowest deviation of the experimental errors percent relative to the expected error percent and the R2AC method, the closest average of the experimental errors percent to the expected error percent.

6. Power laws in preferential attachment graphs and Stein's method for the negative binomial distribution

Ross, Nathan

2013-01-01

For a family of linear preferential attachment graphs, we provide rates of convergence for the total variation distance between the degree of a randomly chosen vertex and an appropriate power law distribution as the number of vertices tends to infinity. Our proof uses a new formulation of Stein's method for the negative binomial distribution, which stems from a distributional transformation that has the negative binomial distributions as the only fixed points.

7. The zero inflated negative binomial – Crack distribution: some properties and parameter estimation

Pornpop Saengthong

2015-12-01

Full Text Available The zero inflated negative binomial-Crack (ZINB-CR distribution is a mixture of Bernoulli distribution and negative binomial-Crack (NB-CR distribution, which is an alternative distribution for the excessive zero counts and overdispersion. In this paper, some properties of the ZINB-CR distribution are discussed. Statistical inference of the parameters is derived by maximum likelihood estimation (MLE and the method of moments (MM. Monte Carlo Simulations are used to evaluate the performance of parameter estimation methods in term of mean squared error (MSE. An application of the distribution is carried out on a sample of excess zero-count data. Simulation results show that the MLE method outperforms the MM method in specific parameter values. Furthermore, the ZINB-CR provides a better fit compared to the zero inflated Poisson (ZIP, the zero inflated negative binomial (ZINB and the negative binomial-Crack (NB-CR distributions.

8. Chromosome aberration analysis based on a beta-binomial distribution

Analyses carried out here generalized on earlier studies of chromosomal aberrations in the populations of Hiroshima and Nagasaki, by allowing extra-binomial variation in aberrant cell counts corresponding to within-subject correlations in cell aberrations. Strong within-subject correlations were detected with corresponding standard errors for the average number of aberrant cells that were often substantially larger than was previously assumed. The extra-binomial variation is accomodated in the analysis in the present report, as described in the section on dose-response models, by using a beta-binomial (B-B) variance structure. It is emphasized that we have generally satisfactory agreement between the observed and the B-B fitted frequencies by city-dose category. The chromosomal aberration data considered here are not extensive enough to allow a precise discrimination between competing dose-response models. A quadratic gamma ray and linear neutron model, however, most closely fits the chromosome data. (author)

9. Analysis of generalized negative binomial distributions attached to hyperbolic Landau levels

Chhaiba, Hassan; Demni, Nizar; Mouayn, Zouhair

2016-01-01

To each hyperbolic Landau level of the Poincar\\'e disc is attached a generalized negative binomial distribution. In this paper, we compute the moment generating function of this distribution and supply its decomposition as a perturbation of the negative binomial distribution by a finitely-supported measure. Using the Mandel parameter, we also discuss the nonclassical nature of the associated coherent states. Next, we determine the L\\'evy-Kintchine decomposition its characteristic function whe...

10. Bacterial Density in Water Determined by Poisson or Negative Binomial Distributions

El-Shaarawi, A. H.; Esterby, S. R.; Dutka, B. J.

1981-01-01

The question of how to characterize the bacterial density in a body of water when data are available as counts from a number of small-volume samples was examined for cases where either the Poisson or negative binomial probability distributions could be used to describe the bacteriological data. The suitability of the Poisson distribution when replicate analyses were performed under carefully controlled conditions and of the negative binomial distribution for samples collected from different l...

11. A COMPOUND OF ZERO TRUNCATED GENERALIZED NEGATIVE BINOMIAL DISTRIBUTION WITH GENERALIZED BETA DISTRIBUTION

2013-01-01

In this paper, we provide a chronological overview of the recent developments in thecompounding of distributions. An attempt has been made to obtain a compound of zero truncatedgeneralized negative binomial distribution (Ztgnbd) with that of Generalized Beta Distribution(GBD). Factorial and ordinary crude moments of some zero truncated compound distributionshave also been discussed. The compound is then specialized by varying the value of β in((Ztgnbd)).

12. A COMPOUND OF ZERO TRUNCATED GENERALIZED NEGATIVE BINOMIAL DISTRIBUTION WITH GENERALIZED BETA DISTRIBUTION

2013-01-01

Full Text Available In this paper, we provide a chronological overview of the recent developments in thecompounding of distributions. An attempt has been made to obtain a compound of zero truncatedgeneralized negative binomial distribution (Ztgnbd with that of Generalized Beta Distribution(GBD. Factorial and ordinary crude moments of some zero truncated compound distributionshave also been discussed. The compound is then specialized by varying the value of β in((Ztgnbd.

13. Characterizations of the Extended Geometric, Harris, Negative Binomial and Gamma Distributions

Sandhya, E.; Sherly, S; Jos, M K; N. Raju

2005-01-01

Extended geometric distribution is defined and its mixture is characterized by the property of having completely monotone probability sequence. Also, convolution equations and probability generating functions are used to characterize extended geometric distributions. Further, some characterizations of Harris and negative binomial distributions based on probability generating functions are obtained. Relations between these distributions are derived and finally a gamma distribution is character...

14. Analyses of multiplicity distributions by means of the Modified Negative Binomial Distribution and its KNO scaling function

OSADA, T; Nakajima, N; Biyajima, M.; Suzuki, N.

1998-01-01

We analyze various data of multiplicity distributions by means of the Modified Negative Binomial Distribution (MNBD) and its KNO scaling function, since this MNBD explains the oscillating behavior of the cumulant moment observed in e^+e^- annihilations, h-h collisions and e-p collisions. In the present analyses, we find that the MNBD(discrete distributions) describes the data of charged particles in e^+e^- annihilations much better than the Negative Binomial Distribution (NBD). To investigate...

15. Analysis of generalized negative binomial distributions attached to hyperbolic Landau levels

Chhaiba, Hassan; Demni, Nizar; Mouayn, Zouhair

2016-07-01

To each hyperbolic Landau level of the Poincaré disc is attached a generalized negative binomial distribution. In this paper, we compute the moment generating function of this distribution and supply its atomic decomposition as a perturbation of the negative binomial distribution by a finitely supported measure. Using the Mandel parameter, we also discuss the nonclassical nature of the associated coherent states. Next, we derive a Lévy-Khintchine-type representation of its characteristic function when the latter does not vanish and deduce that it is quasi-infinitely divisible except for the lowest hyperbolic Landau level corresponding to the negative binomial distribution. By considering the total variation of the obtained quasi-Lévy measure, we introduce a new infinitely divisible distribution for which we derive the characteristic function.

16. Parameter Estimation and Hypothesis Testing of Two Negative Binomial Distribution Population with Missing Data

Zhao, Zhiwen

Our purpose is to deal with the parameter estimation and hypothesis testing on the equality of two negative binomial distribution populations with missing data. The consistency and asymptotic normality of the estimations are proved. In addition statistic on testing equality of two negative distributions and its limiting distribution are obtained.

17. Modified Negative Binomial Description of the Multiplicity Distributions in Lepton-nucleon Scattering

Tchikilev, O. G.

1996-01-01

It is shown that charged hadron multiplicity distributions in lepton-nucleon scattering are fairly well described by the modified negative binomial distribution in the energy range from 3-4 to 220 GeV. The energy behaviour of the parameter k is similar to the dependence observed for e+e- annihilation.

18. Sums of Possibly Associated Bernoulli Variables: The Conway–Maxwell-Binomial Distribution

2016-01-01

The study of sums of possibly associated Bernoulli random variables has been hampered by an asymmetry between positive correlation and negative correlation. The Conway-Maxwell Binomial (COMB) distribution and its multivariate extension, the Conway-Maxwell Multinomial (COMM) distribution, gracefully model both positive and negative association. Sufficient statistics and a family of proper conjugate distributions are found. The relationship of this distribution to the exchangeable special case ...

19. A generalized negative binomial distribution based on an extended Poisson process

Salasar, Luis Ernesto Bueno; Leite, José Galvão; Neto, Francisco Louzada

2010-01-01

In this article we propose a generalized negative binomial distribution, which is constructed based on an extended Poisson process (a generalization of the homogeneous Poisson process). This distribution is intended to model discrete data with presence of zero-inflation and over-dispersion. For a dataset on animal abundance which presents over-dispersion and a high frequency of zeros, a comparison between our extended distribution and other common distributions used for modeling this kind of ...

20. Multiplicity Distributions in Strong Interactions: A Generalized Negative Binomial Model

Hegyi, S.

1996-01-01

A three-parameter discrete distribution is developed to describe the multiplicity distributions observed in total- and limited phase space volumes in different collision processes. The probability law is obtained by the Poisson transform of the KNO scaling function derived in Polyakov's similarity hypothesis for strong interactions as well as in perturbative QCD. Various characteristics of the newly proposed distribution are investigated e.g. its generating function, factorial moments, factor...

1. Multiplicity Distributions in Strong Interactions A Generalized Negative Binomial Model

Hegyi, S

1996-01-01

A three-parameter discrete distribution is developed to describe the multiplicity distributions observed in total- and limited phase space volumes in different collision processes. The probability law is obtained by the Poisson transform of the KNO scaling function derived in Polyakov's similarity hypothesis for strong interactions as well as in perturbative QCD. Various characteristics of the newly proposed distribution are investigated e.g. its generating function, factorial moments, factorial cumulants. Several limiting and special cases are discussed. A comparison is made to the multiplicity data available in e+e- annihilations at the Z0 peak.

2. The Negative Binomial Distribution as a Renewal Model for the Recurrence of Large Earthquakes

Tejedor, Alejandro; Gómez, Javier B.; Pacheco, Amalio F.

2015-01-01

The negative binomial distribution is presented as the waiting time distribution of a cyclic Markov model. This cycle simulates the seismic cycle in a fault. As an example, this model, which can describe recurrences with aperiodicities between 0 and 0.5, is used to fit the Parkfield, California earthquake series in the San Andreas Fault. The performance of the model in the forecasting is expressed in terms of error diagrams and compared with other recurrence models from literature.

3. Binomial Distribution Sample Confidence Interval Estimation for Positive and Negative Likelihood Ratio Medical Key Parameters

Sorana BOLBOACĂ; Lorentz JÄNTSCHI

2005-01-01

Likelihood Ratio medical key parameters calculated on categorical results from diagnostic tests are usually express accompanied with their confidence intervals, computed using the normal distribution approximation of binomial distribution. The approximation creates known anomalies, especially for limit cases. In order to improve the quality of estimation, four new methods (called here RPAC, RPAC0, RPAC1, and RPAC2) were developed and compared with the classical method (called h...

4. Studying the Binomial Distribution Using LabVIEW

George, Danielle J.; Hammer, Nathan I.

2015-01-01

This undergraduate physical chemistry laboratory exercise introduces students to the study of probability distributions both experimentally and using computer simulations. Students perform the classic coin toss experiment individually and then pool all of their data together to study the effect of experimental sample size on the binomial…

5. Estimating the abundance of clustered animal population by using adaptive cluster sampling and negative binomial distribution

Bo, Yizhou; Shifa, Naima

2013-09-01

An estimator for finding the abundance of a rare, clustered and mobile population has been introduced. This model is based on adaptive cluster sampling (ACS) to identify the location of the population and negative binomial distribution to estimate the total in each site. To identify the location of the population we consider both sampling with replacement (WR) and sampling without replacement (WOR). Some mathematical properties of the model are also developed.

6. SERVICE LEVEL ANALYSIS OF (S - 1, S) INVENTORY POLICY FOR NEGATIVE BINOMIAL DISTRIBUTED FAILURES

SUNEUNG AHN; WOOHYUN KIM

2008-01-01

This paper deals with the analysis of uncertain parameters in the (S - 1, S) inventory control policy. In order to determine the number of spare parts under the policy, parameters in a probability model for the number of failures during a replenishment lead time should be interpreted first. In case of the negative binomial distributed failures, the presented interpretation facilitates the incorporation of experts' opinions into the estimation of uncertain parameters. Determination of paramete...

7. Two Different Definitions of Negative Binomial Distributions%负二项分布的两个不同定义

康殿统

2014-01-01

给出了负二项分布的两个不同定义，给出了两类负二项随机变量的期望、方差与矩母函数。从直观上对这两类负二项随机变量做了描述。%Two definitions of the nega tive binomial distributions are introduced. The expectations, variances and moment generating functions for two types of the negative binomial random variables are computed. Intuitive and descriptive explanations are made for the negative binomial random variables.

8. Linking the Negative Binomial and Logarithmic Series Distributions via their Associated Series Relacionando las distribuciones binomial negativa\\ y logarítmica vía sus series asociadas

Full Text Available The negative binomial distribution is associated to the series obtained by taking derivatives of the logarithmic series. Conversely, the logarithmic series distribution is associated to the series found by integrating the series associated to the negative binomial distribution. The parameter of the number of failures of the negative binomial distribution is the number of derivatives needed to obtain the negative binomial series from the logarithmic series. The reasoning in this article could be used as an alternative method to prove that the probability mass function of the negative binomial distribution sums to one. Finally, an interpretation of the logarithmic series distribution is given by using the presented reasoning.La distribución binomial negativa está asociada a la serie obtenida de derivar la serie logarítmica. Recíprocamente, la distribución logarítmica está asociada a la serie obtenida de integrar la serie asociada a la distribución binomial negativa. El parámetro del número de fallas de la distribución binomial negativa es el número de derivadas necesarias para obtener la serie binomial negativa de la serie logarítmica. El razonamiento presentado puede emplearse como un método alternativo para probar que la función de masa de probabilidad de la distribución binomial negativa suma uno. Finalmente, se presenta una interpretación de la distribución logarítmica usando el razonamiento planteado.

9. Multiplicities in ultrarelativistic proton-(anti)proton collisions and negative binomial distribution fits

Prorok, Dariusz

2011-01-01

Likelihood ratio tests are performed for the hypothesis that charged-particle multiplicities measured in proton-(anti)proton collisions at $\\sqrt{s}$ = 0.9 and 2.36 TeV are distributed according to the negative binomial form. Results indicate that the hypothesis should be rejected in the all cases of ALICE-LHC measurements in the limited pseudo-rapidity windows, whereas should be accepted in the corresponding cases of UA5 data. Possible explanations of that and of the disagreement with the le...

10. Multiplicities in ultrarelativistic proton-(anti)proton collisions and negative binomial distribution fits

Prorok, Dariusz

2011-01-01

Likelihood ratio tests are performed for the hypothesis that charged-particle multiplicities measured in proton-(anti)proton collisions at $\\sqrt{s}$ = 0.9, 2.36 TeV are distributed according to the negative binomial form. Results indicate that the hypothesis should be rejected in the all cases of LHC measurements in the limited pseudo-rapidity windows, whereas should be accepted in the corresponding cases of UA5 data. Possible explanations of that and of the disagreement with the least-squares fitting method are given.

11. A comparison of LMC and SDL complexity measures on binomial distributions

Piqueira, José Roberto C.

2016-02-01

The concept of complexity has been widely discussed in the last forty years, with a lot of thinking contributions coming from all areas of the human knowledge, including Philosophy, Linguistics, History, Biology, Physics, Chemistry and many others, with mathematicians trying to give a rigorous view of it. In this sense, thermodynamics meets information theory and, by using the entropy definition, López-Ruiz, Mancini and Calbet proposed a definition for complexity that is referred as LMC measure. Shiner, Davison and Landsberg, by slightly changing the LMC definition, proposed the SDL measure and the both, LMC and SDL, are satisfactory to measure complexity for a lot of problems. Here, SDL and LMC measures are applied to the case of a binomial probability distribution, trying to clarify how the length of the data set implies complexity and how the success probability of the repeated trials determines how complex the whole set is.

12. Energy dependence of parameters of negative-binomial distributions in NN collisions

We have discussed the energy dependence of the parameters left-angle n right-angle and K of the negative-binomial distributions for the full phase-space charged-particle multiplicity distributions in pp/p bar p collisions. It was shown that, under the assumption that multiparticle production is a stationary branching process, there exists a relation between the two parameters left-angle n right-angle and K. A similar relation can also be derived empirically, from information-theoretic entropy considerations. The energy dependence of one parameter then automatically determines the energy dependence of the other. It was then argued from entropy considerations that there should be an upper bound for left-angle n right-angle or conversely a lower bound for K as a function of energy. New parametrizations for the energy dependence of K were given, taking into account that K should have a lower bound, which together with the relation between the parameters left-angle n right-angle and K predicts that the average number of particles cannot be increased indefinitely with increasing c.m. system energy

13. Analysis of DAR(1)/D/s Queue with Quasi-Negative Binomial-II as Marginal Distribution

Kanichukattu Korakutty Jose; Bindu Abraham

2011-01-01

In this paper we consider the arrival process of a multiserver queue governed by a discrete autoregressive process of order 1 [DAR(1)] with Quasi-Negative Binomial Distribution-II as the marginal distribution. This discrete time multiserver queueing system with autoregressive arrivals is more suitable for modeling the Asynchronous Transfer Mode(ATM) multiplexer queue with Variable Bit Rate (VBR) coded teleconference traffic. DAR(1) is described by a few parameters and it is easy to match the ...

14. Design and analysis of three-arm trials with negative binomially distributed endpoints.

Mütze, Tobias; Munk, Axel; Friede, Tim

2016-02-20

A three-arm clinical trial design with an experimental treatment, an active control, and a placebo control, commonly referred to as the gold standard design, enables testing of non-inferiority or superiority of the experimental treatment compared with the active control. In this paper, we propose methods for designing and analyzing three-arm trials with negative binomially distributed endpoints. In particular, we develop a Wald-type test with a restricted maximum-likelihood variance estimator for testing non-inferiority or superiority. For this test, sample size and power formulas as well as optimal sample size allocations will be derived. The performance of the proposed test will be assessed in an extensive simulation study with regard to type I error rate, power, sample size, and sample size allocation. For the purpose of comparison, Wald-type statistics with a sample variance estimator and an unrestricted maximum-likelihood estimator are included in the simulation study. We found that the proposed Wald-type test with a restricted variance estimator performed well across the considered scenarios and is therefore recommended for application in clinical trials. The methods proposed are motivated and illustrated by a recent clinical trial in multiple sclerosis. The R package ThreeArmedTrials, which implements the methods discussed in this paper, is available on CRAN. PMID:26388314

15. Inferences and Power Analysis Concerning Two Negative Binomial Distributions with An Application to MRI Lesion Counts Data

Aban, Inmaculada B.; CUTTER, GARY R.; Mavinga, Nsoki

2008-01-01

In comparing the mean count of two independent samples, some practitioners would use the t-test or the Wilcoxon rank sum test while others may use methods based on a Poisson model. It is not uncommon to encounter count data that exhibit overdispersion where the Poisson model is no longer appropriate. This paper deals with methods for overdispersed data using the negative binomial distribution resulting from a Poisson-Gamma mixture. We investigate the small sample properties of the likelihood-...

16. On the multiplicity distribution in statistical model: (I) negative binomial distribution

Xu, Hao-jie

2016-01-01

With the distribution of principal thermodynamic variables (e.g.,volume) and the probability condition from reference multiplicity, we develop an improved baseline measure for multiplicity distribution in statistical model to replace the traditional Poisson expectations. We demonstrate the mismatches between experimental measurements and previous theoretical calculations on multiplicity distributions. We derive a general expression for multiplicity distribution, i.e. a conditional probability distribution, in statistical model and calculate its cumulants under Poisson approximation in connection with recent data for multiplicity fluctuations. We find that probability condition from reference multiplicity are crucial to explain the centrality resolution effect in experiment. With the improved baseline measure for multiplicity distribution, we can quantitatively reproduce the cumulants (cumulant products) for multiplicity distribution of total (net) charges measured in experiments.

17. Informative prior distributions for a binomial model to predict professional tennis results

Colin, Pierre; Bechler, Aurélien

2015-01-01

Tennis is a sport, as many others, that appears to be quite simple in the type of results (victory of one of the two players) but rather quite complex in factors that leads to this binary outcome. The perpetual evolution and increase of the way to collect data leads to more and more accurate available information about professional tennis matches. We studied the predictive properties of the binomial model representing the victory of one player against the other. Bayesian framework enables the...

18. Compound Negative Binomial Distribution and Compound Poisson Distribution%复合负二项分布与复合泊松分布

魏瑛源

2014-01-01

运用矩母函数证明了任何一个复合负二项分布可以写成一个复合泊松分布，并给出一个具体实例。%By using the moment generating function , we prove that a compound negative binomial distribution can be expressed as a compound Poisson distribution .An example is shown .

19. Negative binomials and multiplicity distributions in 250 GeV/c K + and π+ interactions on Al and Au nuclei

Ajinenko, I. V.; Belokopytov, Yu. A.; Bialkowska, H.; Boettcher, H.; Botterweck, F.; Charlet, M.; Chliapnikov, P. V.; Crijns, F.; de Roeck, A.; de Wolf, D. A.; Dziunikowska, K.; Endler, A. M. F.; Eskreys, A.; Garutchava, Z. C.; Golubkov, Y. A.; Gulkanyan, G. R.; van Hal, P.; Hakobyan, R. Sh.; Haupt, T.; Kittel, W.; Kisielewska, D.; Levchenko, B. B.; Machowski, B.; Meijers, F.; Michałowska, A. B.; Nikolaenko, V. I.; Olkiewicz, K.; Pöllänen, R.; Ronjin, V. M.; Rybin, A. M.; Saarikko, H. M. T.; Scholten, L.; Smirnova, L. N.; Tchikilev, O. G.; Uvarov, V. A.; Verbeure, F.; Wischnewski, R.

1990-12-01

The negative binomial distribution (NBD) is fitted to all charged and to negative particle multiplicity distributions in restricted rapidity intervals, both in the forward and backward c.m. hemispheres of positive meson interactions on Al and Au nuclei. For negative particle multiplicity distributions, the NBD parameters are also determined as a function of n g, the number of grey tracks, corresponding to varying number of intranuclear collisions. The data are interpreted in terms of the clan picture of Giovannini and Van Hove and compared to the MCMHA and Fritiof models. Both models reproduce quite well the global multiplicity distributions, but not when sub-samples are considered with fixed number of grey tracks. Regularities are better visible on the parton than on the particle level.

20. Binomial Distribution Sample Confidence Intervals Estimation 8. Number Needed to Treat/Harm

Sorana BOLBOACĂ

2004-08-01

Full Text Available Nowadays, the number needed to treat became the most important parameter in reporting the treatment effects in clinical trials, from binary outcomes such as “positive” or “negative”. Defined as a reciprocal of the absolute risk reduction, the number needed to treat is the number of patients who need to be treated to prevent one additional adverse even. In medical literature, the number needed to treat is reported usually with its asymptotic confidence intervals, method that is used by the most software packages even if it is knows that is not the best method. The aim of this paper is to introduce three new methods of computing confidence intervals for number needed to treat/harm.Using PHP programming language was implementing the proposed methods and the asymptotic one (called here IADWald. The performance of each method, for different sample sizes (m, n and different values of binomial variables (X, Y were asses using a set of criterions: the upper and lower boundaries; the average and standard deviation of the experimental errors; the deviation of the experimental errors relative to imposed significance level (α = 5%. The methods were assess on random binomial variables X, Y (where X < m, Y < n and random sample sizes m, n (4 ≤ m, n ≤ 1000.The performances of the implemented methods of computing confidence intervals for number needed to treat/harm are present in order to be taking into consideration when a confidence interval for number needed to treat is used.

1. Modeling citrus huanglongbing data using a zero-inflated negative binomial distribution

Eudmar Paiva de Almeida

2016-06-01

Full Text Available Zero-inflated data from field experiments can be problematic, as these data require the use of specific statistical models during the analysis process. This study utilized the zero-inflated negative binomial (ZINB model with the log- and logistic-link functions to describe the incidence of plants with Huanglongbing (HLB, caused by Candidatus liberibacter spp. in commercial citrus orchards in the Northwestern Parana State, Brazil. Each orchard was evaluated at different times. The ZINB model with random effects in both link functions provided the best fit, as the inclusion of these effects accounted for variations between orchards and the numbers of diseased plants. The results of this model show that older plants exhibit a lower probability of acquiring HLB. The application of insecticides on a calendar basis or during new foliage flushes resulted in a three times larger probability of developing HLB compared with applying insecticides only when the vector was detected.

2. Binomial Distribution Sample Confidence Intervals Estimation 7. Absolute Risk Reduction and ARR-like Expressions

2004-08-01

Full Text Available Assessments of a controlled clinical trial suppose to interpret some key parameters as the controlled event rate, experimental event date, relative risk, absolute risk reduction, relative risk reduction, number needed to treat when the effect of the treatment are dichotomous variables. Defined as the difference in the event rate between treatment and control groups, the absolute risk reduction is the parameter that allowed computing the number needed to treat. The absolute risk reduction is compute when the experimental treatment reduces the risk for an undesirable outcome/event. In medical literature when the absolute risk reduction is report with its confidence intervals, the method used is the asymptotic one, even if it is well know that may be inadequate. The aim of this paper is to introduce and assess nine methods of computing confidence intervals for absolute risk reduction and absolute risk reduction – like function.Computer implementations of the methods use the PHP language. Methods comparison uses the experimental errors, the standard deviations, and the deviation relative to the imposed significance level for specified sample sizes. Six methods of computing confidence intervals for absolute risk reduction and absolute risk reduction-like functions were assessed using random binomial variables and random sample sizes.The experiments shows that the ADAC, and ADAC1 methods obtains the best overall performance of computing confidence intervals for absolute risk reduction.

3. First Study of the Negative Binomial Distribution Applied to Higher Moments of Net-charge and Net-proton Multiplicity Distributions

Tarnowsky, Terence J

2013-01-01

A study of the first four moments (mean, variance, skewness, and kurtosis) and their products ($\\kappa\\sigma^{2}$ and $S\\sigma$) of the net-charge and net-proton distributions in Au+Au collisions at $\\sqrt{\\rm s_{NN}}$ = 7.7-200 GeV from HIJING simulations has been carried out. It is seen that a Poisson does not effectively describe the actual distributions of positive and negative particles (or protons and anti-protons). A discrete probability distribution that effectively describes the raw distributions is the negative binomial (or binomial) distribution (NBD/BD). The NBD/BD have been used to characterize particle production in high-energy particle and nuclear physics. Differences between $\\kappa\\sigma^{2}$ and the Poisson assumption of a factor of four (for net-charge) and 40% (for net-protons) can be accounted for by the NBD/BD. This is the first application of the NBD/BD to describe the behavior of the higher moments of net-charge and net-proton distributions in nucleus-nucleus collisions.

4. First study of the negative binomial distribution applied to higher moments of net-charge and net-proton multiplicity distributions

Tarnowsky, Terence J.; Westfall, Gary D.

2013-07-01

A study of the first four moments (mean, variance, skewness, and kurtosis) and their products (κσ2 and Sσ) of the net-charge and net-proton distributions in Au + Au collisions at √{sNN} = 7.7- 200 GeV from HIJING simulations has been carried out. The skewness and kurtosis and the collision volume independent products κσ2 and Sσ have been proposed as sensitive probes for identifying the presence of a QCD critical point. A discrete probability distribution that effectively describes the separate positively and negatively charged particle (or proton and anti-proton) multiplicity distributions is the negative binomial (or binomial) distribution (NBD/BD). The NBD/BD has been used to characterize particle production in high-energy particle and nuclear physics. Their application to the higher moments of the net-charge and net-proton distributions is examined. Differences between κσ2 and a statistical Poisson assumption of a factor of four (for net-charge) and 40% (for net-protons) can be accounted for by the NBD/BD. This is the first application of the properties of the NBD/BD to describe the behavior of the higher moments of net-charge and net-proton distributions in nucleus-nucleus collisions.

5. Fitting the truncated negative binomial distribution to count data. A comparison of estimators, with an application to groundfishesfrom the Mauritanian Exclusive Economic Zone

Manté, Claude; Kidé, Oumar, Saikou; Yao, Anne-Françoise; Mérigot, Bastien

2016-01-01

International audience A frequent issue in the study of species abundance consists in modeling empirical distributions of repeated counts by parametric probability distributions. In this setting, it is desirable that the chosen family of distributions is exible enough to take into account very diverse patterns, and that its parameters possess clear biological/ecological meanings. This is the case of the Negative Binomial distribution, chosen in this work for modeling counts of marine shes ...

6. Negative binomial multiplicity distribution in proton-proton collisions in limited pseudorapidity intervals at LHC up to sqrt (s) = 7 TeV and the clan model

Ghosh, Premomoy

2012-01-01

Experiments at the Large Hadron Collider (LHC) have measured multiplicity distributions in proton-proton collisions at a new domain of center-of-mass energy ($\\sqrt {s}$) in limited pseudorapidity intervals. We analyze multiplicity distribution data of proton-proton collisions at LHC energies as measured by the Compact Muon Solenoid (CMS) experiment in terms of characteristic parameters of the Negative Binomial Distribution (NBD) function that has played a significant role in describing multi...

7. Binomial probability distribution model-based protein identification algorithm for tandem mass spectrometry utilizing peak intensity information.

Xiao, Chuan-Le; Chen, Xiao-Zhou; Du, Yang-Li; Sun, Xuesong; Zhang, Gong; He, Qing-Yu

2013-01-01

Mass spectrometry has become one of the most important technologies in proteomic analysis. Tandem mass spectrometry (LC-MS/MS) is a major tool for the analysis of peptide mixtures from protein samples. The key step of MS data processing is the identification of peptides from experimental spectra by searching public sequence databases. Although a number of algorithms to identify peptides from MS/MS data have been already proposed, e.g. Sequest, OMSSA, X!Tandem, Mascot, etc., they are mainly based on statistical models considering only peak-matches between experimental and theoretical spectra, but not peak intensity information. Moreover, different algorithms gave different results from the same MS data, implying their probable incompleteness and questionable reproducibility. We developed a novel peptide identification algorithm, ProVerB, based on a binomial probability distribution model of protein tandem mass spectrometry combined with a new scoring function, making full use of peak intensity information and, thus, enhancing the ability of identification. Compared with Mascot, Sequest, and SQID, ProVerB identified significantly more peptides from LC-MS/MS data sets than the current algorithms at 1% False Discovery Rate (FDR) and provided more confident peptide identifications. ProVerB is also compatible with various platforms and experimental data sets, showing its robustness and versatility. The open-source program ProVerB is available at http://bioinformatics.jnu.edu.cn/software/proverb/ . PMID:23163785

8. Use of negative binomial distribution to describe the presence of Anisakis in Thyrsites atun Uso de distribuição binomial negativa para descrever a presença de Anisakis em Thyrsites atun

Patricio Peña-Rehbein

2012-03-01

Full Text Available Nematodes of the genus Anisakis have marine fishes as intermediate hosts. One of these hosts is Thyrsites atun, an important fishery resource in Chile between 38 and 41° S. This paper describes the frequency and number of Anisakis nematodes in the internal organs of Thyrsites atun. An analysis based on spatial distribution models showed that the parasites tend to be clustered. The variation in the number of parasites per host could be described by the negative binomial distribution. The maximum observed number of parasites was nine parasites per host. The environmental and zoonotic aspects of the study are also discussed.Nematóides do gênero Anisakis têm nos peixes marinhos seus hospedeiros intermediários. Um desses hospedeiros é Thyrsites atun, um importante recurso pesqueiro no Chile entre 38 e 41° S. Este artigo descreve a freqüência e o número de nematóides Anisakis nos órgãos internos de Thyrsites atun. Uma análise baseada em modelos de distribuição espacial demonstrou que os parasitos tendem a ficar agrupados. A variação numérica de parasitas por hospedeiro pôde ser descrita por distribuição binomial negativa. O número máximo observado de parasitas por hospedeiro foi nove. Os aspectos ambientais e zoonóticos desse estudo também serão discutidos.

9. An Approximate Interval Estimate on the Parameter of Negative Binomial Distribution%负二项分布参数的一类近似区间估计

姜培华

2012-01-01

借助负二项分布和卡方分布的极限关系，推导给出当参数P较小条件下的近似区间估计，并通过数值例子介绍了此区间估计方法的应用．%An approximate interval estimate for small was gained with the limit relationship of negative binomi- al distribution and the chi - square distribution ; Finally, the method of interval estimate was studied and illustration was shown by examples.

10. Distribusi Markov-Binomial Negatif

Widyasari, Rina

2015-01-01

The way to find a new distribution of random variables is defining the distribution which associated with Markov chain. In this research, researcher defines all the random variables identically independent distributed negative binomial distribution and form a Markov chain. Suppose that Xn is a sequence of Bernoulli trials that if 1 occurs means ”success” and 0 occurs means ”failure”. Nb(s) defined as random variables sth success in n trials. Each trial form a Markov chain, in n...

11. Analyses of multiplicity distributions of e^+e^- and e-p collisions by means of modified negative binomial distribution and Laguerre-type distribution: Interrelation of solutions in stochastic processes

Biyajima, M.; OSADA, T; Takei, K.

1998-01-01

A pure birth stochastic process with several initial conditions is considered.We analyze multiplicity distributions of e^+e^- collisions and e-p collisions, usigthe Modified Negative Binomial Distribution (MNBD) and the Laguerre-type distribution. Several multiplicity distributions show the same minimum \\chi^2's values in analyses by means of two formulas: In these cases, we find that a parameter N contained in the MNBD becomes to be large. Taking large N limit in the MNBD, we find that the L...

12. Negative binomial multiplicity distribution in proton-proton collisions in limited pseudorapidity intervals at LHC up to sqrt (s) = 7 TeV and the clan model

Ghosh, Premomoy

2012-01-01

Experiments at the Large Hadron Collider (LHC) have measured multiplicity distributions in proton-proton collisions at a new domain of center-of-mass energy ($\\sqrt {s}$) in limited pseudorapidity intervals. We analyze multiplicity distribution data of proton-proton collisions at LHC energies as measured by the Compact Muon Solenoid (CMS) experiment in terms of characteristic parameters of the Negative Binomial Distribution (NBD) function that has played a significant role in describing multiplicity distribution data of particle production in high energy physics experiments, in the pre-LHC energy-range, in various kinds of collisions for a wide range of collision energy and for different kinematic ranges. Beside a single NBD, we apply the formalism of weighted superposition of two NBDs to examine if the multiplicity distribution data of CMS could be better explained. The weighted superposition of two NBDs indeed explain the distribution data better at the highest available LHC energy and in large interval of ...

13. Generalized Binomial Trees

Jackwerth, Jens Carsten

1996-01-01

We consider the problem of consistently pricing new options given the prices of related options on the same stock. The Black-Scholes formula and standard binomial trees can only accommodate one related European option which then effectively specifies the volatility parameter. Implied binomial trees can accommodate only related European options with the same time-to-expiration.The generalized binomial trees introduced here can accommodate any kind of related options (European, American, or exo...

14. Properties of Negative Binomial Distribution and Its Applications in the RiskManagement%负二项分布的优良特性及其在风险管理中的应用

王丙参; 何万生; 戴宁

2011-01-01

This article discusses the properties and promotion of the two basic negative binomial distributions,gives closed of conditional probabilities and a non-classical confidence interval estimate under the first negative binomial distribution,discusses the relationship between the second negative binomial distribution to poisson distribution.%研究了负二项分布的两个基本模型及推广,得到第一类负二项分布条件概率具有封闭性且给出参数的一个非经典置信区间估计,特别研究了第二类负二项分布与泊松分布的关系。

15. Negative Binomial States of Quantized Radiation Fields

Fu, Hong-Chen; Sasaki, Ryu

1996-01-01

We introduce the negative binomial states with negative binomial distribution as their photon number distribution. They reduce to the ordinary coherent states and Susskind-Glogower phase states in different limits. The ladder and displacement operator formalisms are found and they are essentially the Perelomov's su(1,1) coherent states via its Holstein-Primakoff realisation. These states exhibit strong squeezing effect and they obey the super-Poissonian statistics. A method to generate these ...

16. Negative Binomial States of Quantized Radiation Fields

Fu, H C; Fu, Hong-Chen; Sasaki, Ryu

1996-01-01

We introduce the negative binomial states with negative binomial distribution as their photon number distribution. They reduce to the ordinary coherent states and Susskind-Glogower phase states in different limits. The ladder and displacement operator formalisms are found and they are essentially the Perelomov's su(1,1) coherent states via its Holstein-Primakoff realisation. These states exhibit strong squeezing effect and they obey the super-Poissonian statistics. A method to generate these states is proposed.

17. Negative binomial multiplicity distribution in proton-proton collisions in limited pseudorapidity intervals at LHC up to s=7TeV and the clan model

Ghosh, Premomoy

2012-03-01

Experiments at the Large Hadron Collider (LHC) have measured multiplicity distributions in proton-proton collisions at a new domain of center-of-mass energy (s) in limited pseudorapidity intervals. We analyze multiplicity distribution data of proton-proton collisions at LHC energies as measured by the Compact Muon Solenoid (CMS) experiment in terms of characteristic parameters of the negative binomial distribution (NBD) function that has played a significant role in describing multiplicity distribution data of particle production in high-energy physics experiments, in the pre-LHC energy range, in various kinds of collisions for a wide range of collision energy and for different kinematic ranges. Beside a single NBD, we apply the formalism of weighted superposition of two NBDs to examine if the multiplicity distribution data of CMS could be better explained. The weighted superposition of two NBDs indeed explains the distribution data better at the highest available LHC energy and in large intervals of phase space. The two-NBD formalism further reveals that the energy invariance of the multiplicity distribution of the soft component of particle production in hadronic collisions is valid at LHC also, as it is at RHIC and Tevatron. We analyze the data further in terms of clan parameters in the framework of the two-NBD model.

18. Two Interval Estimates on the Parameter of Negative Binomial Distribution%负二项分布参数的两种区间估计

姜培华; 范国良

2012-01-01

研究了负二项分布参数的区间估计方法,给出其两种区间估计方法.首先给出负二项分布参数的精确区间估计方法；其次给出大样本近似区间估计方法.最后通过数值例子介绍这些区间估计方法的应用.%The methods of interval estimate on the parameter of negative binomial distribution have been studied. Two methods are given. Firstly, the accurate interval estimate is put forward ; secondly, large sample approximate inter-val estimate is gained; Finally,these methods of interval estimate are studied and illustration is shown by examples.

19. 基于β-二项分布的结构易损性分析%Structural fragility estimation with beta-binomial distribution

刘骁骁; 吴子燕; 王其昂

2014-01-01

易损性曲线建立过程中受激励不确定性和结构参数不确定性的影响，会引起结构或构件观测结果的统计相关性。为此，本文提出基于β-二项分布的结构易损性分析方法。该方法根据性能量化指标阈值和 Monte Carlo模拟确定震后观测结果，采用β-二项分布探讨震后观测值的统计相关性；结合对数回归模型，推导了改进β-二项分布的累积分布函数，计算结构失效概率；通过累积对数正态分布拟合易损性曲线，比较了观测失效样本数与观测失效概率统计相关性对易损性的影响，并与未考虑统计相关性的传统易损性曲线作对比。某8层钢筋混凝土框架-剪力墙结构的算例表明，考虑统计相关性的易损性较传统易损性偏大，且结构遭受8度以上地震作用时，考虑失效样本数统计相关性的易损性使预测结果更为保守，利于工程安全。%The uncertainty of seismic excitation and structural parameters in the process of establishing fragility curves leads to statistical dependence among observations,which has been neglected in past applications.In this paper,a new methodology based on beta-binomial distribution to calculate structural fragility is presented.Observations indicating the states (failure or survival)are confirmed via quantita-tive indicators threshold as well as Monte Carlo after each earthquake.Beta-Binomial distribution is ad-dressed to discuss the statistical dependence among observations.Improved cumulative beta-binomial dis-tribution function is derivation to calculate failure probability combined with logistic regression model. Seismic vulnerability curve can be fitted by means of cumulative lognormal distribution,which is com-pared with traditional fragility that of neglecting statistical dependence among observations and fragility curve considering statistical dependence among observed failure rates.A seat eight floors reinforced con

20. Option pricing for the transformed‐binomial class

António Câmara; San‐Lin Chung

2006-01-01

This article generalizes the seminal Cox‐Ross‐Rubinstein (1979) binomial option pricing model to all members of the class of transformed‐binomial pricing processes. The investigation addresses issues related with asset pricing modeling, hedging strategies, and option pricing. Formulas are derived for (a) replicating or hedging portfolios, (b) risk‐neutral transformed‐binomial probabilities, (c) limiting transformed‐normal distributions, and (d) the value of contingent claims, including limiti...

1. MEASUREMENT ERROR EFFECT ON THE POWER OF THE CONTROL CHART FOR ZERO-TRUNCATED BINOMIAL DISTRIBUTION UNDER STANDARDIZATION PROCEDURE

Anwer Khurshid

2014-12-01

Full Text Available Measurement error effect on the power of control charts for zero truncated Poisson distribution and ratio of two Poisson distributions are recently studied by Chakraborty and Khurshid (2013a and Chakraborty and Khurshid (2013b respectively. In this paper, in addition to the expression for the power of control chart for ZTBD based on standardized normal variate is obtained, numerical calculations are presented to see the effect of errors on the power curve. To study the sensitivity of the monitoring procedure, average run length (ARL is also considered.

2. Negative binomial distribution fits to multiplicity distributions in restricted δη intervals from central 16O + Cu collisions at 14.6A GeV/c and their implication for ''intermittency''

Fluctuations in pseudorapidity of charged particles from central (ZCAL) collisions of 16O + Cu at 14.6 A·GeV/c have been analyzed by the E802 Collaboration using the method of normalized factorial moments as a function of the interval Δη. In agreement with previous measurements, an apparent power-law growth of moments with decreasing interval is observed down to Δη ∼ 0.1. Experience with ET distributions suggests that fluctuations of multiplicity and transverse energy can be well described by Gamma or Negative Binomial Distributions (NBD) and excellent fits to NBD were obtained in all Δη bins. The κ parameter of the NBD fit was found to increase linearly with the Δη interval, which due to the well known property of the NBD under convolution, indicates that the multiplicity distributions in adjacent bins of pseudorapidity Δη ∼ 0.1 are largely statistically independent

3. Sub-binomial light

Sperling, J.; Vogel, W; Agarwal, G. S.

2012-01-01

The click statistics from on-off detector systems is quite different from the counting statistics of the more traditional detectors. This necessitates introduction of new parameters to characterize the nonclassicality of fields from measurements using on-off detectors. To properly replace the Mandel Q_M parameter, we introduce a parameter Q_B. A negative value represents a sub-binomial statistics. This is possible only for quantum fields, even for super-Poisson light. It eliminates the proble...

4. Real and imaginary negative binomial states

Liao, Jing; Wang, Xiaoguang; Wu, Ling-An; Pan, Shao-Hua

2001-10-01

The real and imaginary negative binomial states formed by a superposition of the negative binomial states are introduced. The sub-Poissonian statistics, Wigner function and squeezing properties of the real and imaginary states are studied in detail. The oscillatory character of the photon distribution due to the quantum interference between the two components is shown. Moreover, we find that these states are real and imaginary nonlinear Schrödinger cat states and give the corresponding ladder operator formalisms. We also discuss how to generate these general real quantum superposition states based on the intensity-dependent Jaynes-Cummings model.

5. On a fractional binomial process

Cahoy, Dexter O.; Polito, Federico

2013-01-01

The classical binomial process has been studied by \\citet{jakeman} (and the references therein) and has been used to characterize a series of radiation states in quantum optics. In particular, he studied a classical birth-death process where the chance of birth is proportional to the difference between a larger fixed number and the number of individuals present. It is shown that at large times, an equilibrium is reached which follows a binomial process. In this paper, the classical binomial p...

6. Augment-and-Conquer Negative Binomial Processes

Zhou, Mingyuan; Carin, Lawrence

2012-01-01

By developing data augmentation methods unique to the negative binomial (NB) distribution, we unite seemingly disjoint count and mixture models under the NB process framework. We develop fundamental properties of the models and derive efficient Gibbs sampling inference. We show that the gamma-NB process can be reduced to the hierarchical Dirichlet process with normalization, highlighting its unique theoretical, structural and computational advantages. A variety of NB processes with distinct s...

7. Fourier Spectra of Binomial APN Functions

Bracken, Carl; Markin, Nadya; McGuire, Gary

2008-01-01

In this paper we compute the Fourier spectra of some recently discovered binomial APN functions. One consequence of this is the determination of the nonlinearity of the functions, which measures their resistance to linear cryptanalysis. Another consequence is that certain error-correcting codes related to these functions have the same weight distribution as the 2-error-correcting BCH code. Furthermore, for fields of odd degree, our results provide an alternative proof of the APN property of the functions.

8. Stein factors for negative binomial approximation in Wasserstein distance

Barbour, A. D.; Gan, H.L.; Xia, A

2015-01-01

The paper gives the bounds on the solutions to a Stein equation for the negative binomial distribution that are needed for approximation in terms of the Wasserstein metric. The proofs are probabilistic, and follow the approach introduced in Barbour and Xia (Bernoulli 12 (2006) 943-954). The bounds are used to quantify the accuracy of negative binomial approximation to parasite counts in hosts. Since the infectivity of a population can be expected to be proportional to its total parasite burde...

9. Binomial expansions modulo prime powers

Paul W. Haggard

1980-01-01

Full Text Available In this note a result is given and proved concerning binomial expansions modulo prime powers. In the proof congruence modulo prime powers is generalized to the rational numbers via valuations.

10. A new bivariate negative binomial regression model

Faroughi, Pouya; Ismail, Noriszura

2014-12-01

This paper introduces a new form of bivariate negative binomial (BNB-1) regression which can be fitted to bivariate and correlated count data with covariates. The BNB regression discussed in this study can be fitted to bivariate and overdispersed count data with positive, zero or negative correlations. The joint p.m.f. of the BNB1 distribution is derived from the product of two negative binomial marginals with a multiplicative factor parameter. Several testing methods were used to check overdispersion and goodness-of-fit of the model. Application of BNB-1 regression is illustrated on Malaysian motor insurance dataset. The results indicated that BNB-1 regression has better fit than bivariate Poisson and BNB-2 models with regards to Akaike information criterion.

11. Biased binomial assessment of cross-validated estimation of classification accuracies illustrated in diagnosis predictions

Quentin Noirhomme; Damien Lesenfants; Francisco Gomez; Andrea Soddu; Jessica Schrouff; Gaëtan Garraux; André Luxen; Christophe Phillips; Steven Laureys

2014-01-01

Multivariate classification is used in neuroimaging studies to infer brain activation or in medical applications to infer diagnosis. Their results are often assessed through either a binomial or a permutation test. Here, we simulated classification results of generated random data to assess the influence of the cross-validation scheme on the significance of results. Distributions built from classification of random data with crossvalidation did not follow the binomial distribution. The binomi...

12. 一类新的损失函数下负二项分布模型的Bayes可靠生分析%The Bayesian Analysis for Negative Binomial Distribution under a New Loss Function

王琪; 李玮

2011-01-01

The Bayesian and hierarchical Bayesian estimators of reliability of negative binomial distribution are obtained, and the E Bayesian estimator is also discussed by E Bayesian approach. Finally, a numberical example is given to compare the former three estmators, and the contusion shows that the E Bayesian estimator is better than the hierarchical Bayesian estimator.%在一类新的加权平方损失函数下,给出了负二项分布可靠度的Bayes和多层Bayes估计,并利用EBayes方法给出了负二项分布可靠度的E Bayes估计.针对得到的3种估计,给出了数值模拟,结果表明:EBayes估计比多层Bayes估计优良.

13. Edgeworth Binomial Trees.

Mark Rubinstein.

1997-01-01

This paper develops a simple technique for valuing European and American derivatives with underlying asset risk-neutral returns which depart from lognormal in terms of prespecified non-zero skewness and greater-than-three kurtosis. Instead of specifying the entire risk-neutral distribution by the riskless return and volatility (as in the Black-Scholes case), this distribution is specified by its third and fourth central moments as well. An Edgeworth expansion is used to transform a standard b...

14. Transform methods for testing the negative binomial hypothesis

Simos G. Meintanis

2007-10-01

Full Text Available We employ the empirical probability generating function in constructing a goodness-of-fit test for negative binomial distributions. The proposed tests, which are formed as weighted integrals, are shown to be consistent and their asymptotic null distribution is investigated. As the decay of the weight function tends to infinity, limit statistics are obtained. A small simulation study is presented.

15. Transform methods for testing the negative binomial hypothesis

Meintanis, Simos G.

2007-01-01

We employ the empirical probability generating function in constructing a goodness-of-fit test for negative binomial distributions. The proposed tests, which are formed as weighted integrals, are shown to be consistent and their asymptotic null distribution is investigated. As the decay of the weight function tends to infinity, limit statistics are obtained. A small simulation study is presented.

16. 负二项分布参数的贝叶斯区间估计问题%Research of the Bayesian Interval Estimate on the Parameter of Negative Binomial Distribution

姜培华; 纪习习; 吴玲

2014-01-01

In terms of prior distribution of Beta distribution, the Bayesian estimation method on the unknown parame-ter θ of negative binomial distribution was studied. By means of the relations between Beta distribution and the F dis-tribution the general posterior interval estimation of parameter θ was given, and the shortest posterior interval estima-tion by means of conditional extreme was gained. By comparing the discussion analysis and numerical examples den-sity curve shape of the different parameters, it was concluded that in the case of small samples, the shortest confi-dence interval estimation method is worth using.%研究了在先验分布为贝塔分布下，负二项分布未知参数θ的贝叶斯区间估计方法。借助Beta分布与F分布的关系给出了参数θ的一般后验区间估计，并给出了参数θ的最短后验区间估计的条件极值解法。通过对参数取值不同的密度曲线形状的讨论分析和数值实例对比，得出结论：在小样本情况下，最短置信区间估计方法值得采用。

17. Zero inflated negative binomial-generalized exponential distributionand its applications

Sirinapa Aryuyuen; Winai Bodhisuwan; Thidaporn Supapakorn

2014-01-01

In this paper, we propose a new zero inflated distribution, namely, the zero inflated negative binomial-generalized exponential (ZINB-GE) distribution. The new distribution is used for count data with extra zeros and is an alternative for data analysis with over-dispersed count data. Some characteristics of the distribution are given, such as mean, variance, skewness, and kurtosis. Parameter estimation of the ZINB-GE distribution uses maximum likelihood estimation (MLE) method. Simul...

18. Non-Commutative Q-Binomial Formula

Nalci, Sengul; Pashaev, Oktay

2012-01-01

In this paper, we found new q-binomial formula for Q-commutative operators. Expansion coefficients in this formula are given by q-binomial coefficients with two bases (q,Q), determined by Q-commutative q-Pascal triangle. Our formula generalizes all well-known binomial formulas in the form of Newton, Gauss, symmetrical, non-commutative and Binet-Fibonacci binomials. By our non-commutative q-binomial, we introduce q-analogue of function of two non-commutative variables, which could be used in s...

19. Zero inflated negative binomial-generalized exponential distributionand its applications

Sirinapa Aryuyuen

2014-08-01

Full Text Available In this paper, we propose a new zero inflated distribution, namely, the zero inflated negative binomial-generalized exponential (ZINB-GE distribution. The new distribution is used for count data with extra zeros and is an alternative for data analysis with over-dispersed count data. Some characteristics of the distribution are given, such as mean, variance, skewness, and kurtosis. Parameter estimation of the ZINB-GE distribution uses maximum likelihood estimation (MLE method. Simulated and observed data are employed to examine this distribution. The results show that the MLE method seems to have high efficiency for large sample sizes. Moreover, the mean square error of parameter estimation is increased when the zero proportion is higher. For the real data sets, this new zero inflated distribution provides a better fit than the zero inflated Poisson and zero inflated negative binomial distributions.

20. Numerics of implied binomial trees

Härdle, Wolfgang Karl; Myšičková, Alena

2008-01-01

Market option prices in last 20 years confirmed deviations from the Black and Scholes (BS) models assumptions, especially on the BS implied volatility. Implied binomial trees (IBT) models capture the variations of the implied volatility known as \\volatility smile". They provide a discrete approximation to the continuous risk neutral process for the underlying assets. In this paper, we describe the numerical construction of IBTs by Derman and Kani (DK) and an alternative method by Barle and Ca...

1. Negative Binomial GAM and GAMM to Analyse Amphibian Roadkills

Zuur, Alain; Mira, António; Carvalho, Filipe; Ieno, Elena; Saveliev, A.A.; Smith, G.M.; Walker, N. J.

2009-01-01

This chapter analyses amphibian fatalities along a road in Portugal. The data are counts of kills making a Gaussian distribution unlikely; restricting our choice of techniques. We began with generalised linear models (GLM) and generalised addi- tive models (GAM) with a Poisson distribution, but these models were overdis- persed. To solve this, you can either apply a quasi-Poisson GLM or GAM, or use the negative binomial distribution (Chapter 9). In this particular example, eith...

2. Generalized Negative Binomial Processes and the Representation of Cluster Structures

Zhou, Mingyuan

2013-01-01

The paper introduces the concept of a cluster structure to define a joint distribution of the sample size and its exchangeable random partitions. The cluster structure allows the probability distribution of the random partitions of a subset of the sample to be dependent on the sample size, a feature not presented in a partition structure. A generalized negative binomial process count-mixture model is proposed to generate a cluster structure, where in the prior the number of clusters is finite...

3. Negative Binomial Process Count and Mixture Modeling.

Zhou, Mingyuan; Carin, Lawrence

2015-02-01

The seemingly disjoint problems of count and mixture modeling are united under the negative binomial (NB) process. A gamma process is employed to model the rate measure of a Poisson process, whose normalization provides a random probability measure for mixture modeling and whose marginalization leads to an NB process for count modeling. A draw from the NB process consists of a Poisson distributed finite number of distinct atoms, each of which is associated with a logarithmic distributed number of data samples. We reveal relationships between various count- and mixture-modeling distributions and construct a Poisson-logarithmic bivariate distribution that connects the NB and Chinese restaurant table distributions. Fundamental properties of the models are developed, and we derive efficient Bayesian inference. It is shown that with augmentation and normalization, the NB process and gamma-NB process can be reduced to the Dirichlet process and hierarchical Dirichlet process, respectively. These relationships highlight theoretical, structural, and computational advantages of the NB process. A variety of NB processes, including the beta-geometric, beta-NB, marked-beta-NB, marked-gamma-NB and zero-inflated-NB processes, with distinct sharing mechanisms, are also constructed. These models are applied to topic modeling, with connections made to existing algorithms under Poisson factor analysis. Example results show the importance of inferring both the NB dispersion and probability parameters. PMID:26353243

4. A Class of Binomial Permutation Polynomials

Tu, Ziran; Zeng, Xiangyong; Hu, Lei; Li, Chunlei

2013-01-01

In this note, a criterion for a class of binomials to be permutation polynomials is proposed. As a consequence, many classes of binomial permutation polynomials and monomial complete permutation polynomials are obtained. The exponents in these monomials are of Niho type.

5. Priors for Random Count Matrices Derived from a Family of Negative Binomial Processes

2014-01-01

We define a family of probability distributions for random count matrices with a potentially unbounded number of rows and columns. The three distributions we consider are derived from the gamma-Poisson, gamma-negative binomial, and beta-negative binomial processes. Because the models lead to closed-form Gibbs sampling update equations, they are natural candidates for nonparametric Bayesian priors over count matrices. A key aspect of our analysis is the recognition that, although the random co...

6. Stochastic analysis of complex reaction networks using binomial moment equations.

Barzel, Baruch; Biham, Ofer

2012-09-01

The stochastic analysis of complex reaction networks is a difficult problem because the number of microscopic states in such systems increases exponentially with the number of reactive species. Direct integration of the master equation is thus infeasible and is most often replaced by Monte Carlo simulations. While Monte Carlo simulations are a highly effective tool, equation-based formulations are more amenable to analytical treatment and may provide deeper insight into the dynamics of the network. Here, we present a highly efficient equation-based method for the analysis of stochastic reaction networks. The method is based on the recently introduced binomial moment equations [Barzel and Biham, Phys. Rev. Lett. 106, 150602 (2011)]. The binomial moments are linear combinations of the ordinary moments of the probability distribution function of the population sizes of the interacting species. They capture the essential combinatorics of the reaction processes reflecting their stoichiometric structure. This leads to a simple and transparent form of the equations, and allows a highly efficient and surprisingly simple truncation scheme. Unlike ordinary moment equations, in which the inclusion of high order moments is prohibitively complicated, the binomial moment equations can be easily constructed up to any desired order. The result is a set of equations that enables the stochastic analysis of complex reaction networks under a broad range of conditions. The number of equations is dramatically reduced from the exponential proliferation of the master equation to a polynomial (and often quadratic) dependence on the number of reactive species in the binomial moment equations. The aim of this paper is twofold: to present a complete derivation of the binomial moment equations; to demonstrate the applicability of the moment equations for a representative set of example networks, in which stochastic effects play an important role. PMID:23030885

7. Combinatorial clustering and the beta negative binomial process

Broderick, Tamara; Mackey, Lester; Paisley, John; Jordan, Michael I

2011-01-01

We develop a Bayesian nonparametric approach to a general family of latent class problems in which individuals can belong simultaneously to multiple classes and where each class can be exhibited multiple times by an individual. We introduce a combinatorial stochastic process known as the negative binomial process (NBP) as an infinite-dimensional prior appropriate for such problems. We show that the NBP is conjugate to the beta process, and we characterize the posterior distribution under the ...

8. Sampling from a couple of positively correlated binomial variables

Catalani, Mario

2002-01-01

We know that the marginals in a multinomial distribution are binomial variates exhibiting a negative correlation. But we can construct two linear combinations of such marginals in such a way to obtain a positive correlation. We discuss the restrictions that are to be imposed on the parameters of the given marginals to accomplish such a result. Next we discuss the regression function, showing that it is a linear function but not homoscedastic.

9. A Note on Bayesian Estimation for the Negative-Binomial Model

L. Lio, Y.

2009-01-01

2000 Mathematics Subject Classification: 62F15. The Negative Binomial model, which is generated by a simple mixture model, has been widely applied in the social, health and economic market prediction. The most commonly used methods were the maximum likelihood estimate (MLE) and the moment method estimate (MME). Bradlow et al. (2002) proposed a Bayesian inference with beta-prime and Pearson Type VI as priors for the negative binomial distribution. It is due to the complicated posterior dens...

10. Higher Order Asymptotics for Negative Binomial Regression Inferences from RNA-Sequencing Data

Di, Yanming; Emerson, Sarah C.; Daniel W Schafer; Jeffrey A Kimbrel; Chang, Jeff H

2013-01-01

RNA sequencing (RNA-Seq) is the current method of choice for characterizing transcriptomes and quantifying gene expression changes. This next generation sequencing-based method provides unprecedented depth and resolution. The negative binomial (NB) probability distribution has been shown to be a useful model for frequencies of mapped RNA-Seq reads and consequently provides a basis for statistical analysis of gene expression. Negative binomial exact tests are available for two-g...

11. Newton Binomial Formulas in Schubert Calculus

Cordovez, Jorge; Gatto, Letterio; Santiago, Taise

2008-01-01

We prove Newton's binomial formulas for Schubert Calculus to determine numbers of base point free linear series on the projective line with prescribed ramification divisor supported at given distinct points.

12. Nonparametric Bayesian Negative Binomial Factor Analysis

Zhou, Mingyuan

2016-01-01

A common approach to analyze an attribute-instance count matrix, an element of which represents how many times an attribute appears in an instance, is to factorize it under the Poisson likelihood. We show its limitation in capturing the tendency for an attribute present in an instance to both repeat itself and excite related ones. To address this limitation, we construct negative binomial factor analysis (NBFA) to factorize the matrix under the negative binomial likelihood, and relate it to a...

13. Computation of Greeks using Binomial Tree

Yoshifumi Muroi; Shintaro Suda

2014-01-01

This paper proposes a new efficient algorithm for the computation of Greeks for options using the binomial tree. We also show that Greeks for European options introduced in this article are asymptotically equivalent to the discrete version of Malliavin Greeks. This fact enables us to show that our Greeks converge to Malliavin Greeks in the continuous time model. The computation algorithms of Greeks for American options using the binomial tree is also given in this article. There are three adv...

14. A mixed time series model of binomial counts

Khoo, Wooi Chen; Ong, Seng Huat

2015-10-01

Continuous time series modelling has been an active research in the past few decades. However, time series data in terms of correlated counts appear in many situations such as the counts of rainy days and access downloading. Therefore, the study on count data has become popular in time series modelling recently. This article introduces a new mixture model, which is an univariate non-negative stationary time series model with binomial marginal distribution, arising from the combination of the well-known binomial thinning and Pegram's operators. A brief review of important properties will be carried out and the EM algorithm is applied in parameter estimation. A numerical study is presented to show the performance of the model. Finally, a potential real application will be presented to illustrate the advantage of the new mixture model.

15. Exact Group Sequential Methods for Estimating a Binomial Proportion

Zhengjia Chen

2013-01-01

Full Text Available We first review existing sequential methods for estimating a binomial proportion. Afterward, we propose a new family of group sequential sampling schemes for estimating a binomial proportion with prescribed margin of error and confidence level. In particular, we establish the uniform controllability of coverage probability and the asymptotic optimality for such a family of sampling schemes. Our theoretical results establish the possibility that the parameters of this family of sampling schemes can be determined so that the prescribed level of confidence is guaranteed with little waste of samples. Analytic bounds for the cumulative distribution functions and expectations of sample numbers are derived. Moreover, we discuss the inherent connection of various sampling schemes. Numerical issues are addressed for improving the accuracy and efficiency of computation. Computational experiments are conducted for comparing sampling schemes. Illustrative examples are given for applications in clinical trials.

16. PENERAPAN REGRESI BINOMIAL NEGATIF UNTUK MENGATASI OVERDISPERSI PADA REGRESI POISSON

2013-09-01

Full Text Available Poisson regression was used to analyze the count data which Poisson distributed. Poisson regression analysis requires state equidispersion, in which the mean value of the response variable is equal to the value of the variance. However, there are deviations in which the value of the response variable variance is greater than the mean. This is called overdispersion. If overdispersion happens and Poisson Regression analysis is being used, then underestimated standard errors will be obtained. Negative Binomial Regression can handle overdispersion because it contains a dispersion parameter. From the simulation data which experienced overdispersion in the Poisson Regression model it was found that the Negative Binomial Regression was better than the Poisson Regression model.

17. Biased binomial assessment of cross-validated estimation of classification accuracies illustrated in diagnosis predictions

Quentin Noirhomme

2014-01-01

Full Text Available Multivariate classification is used in neuroimaging studies to infer brain activation or in medical applications to infer diagnosis. Their results are often assessed through either a binomial or a permutation test. Here, we simulated classification results of generated random data to assess the influence of the cross-validation scheme on the significance of results. Distributions built from classification of random data with cross-validation did not follow the binomial distribution. The binomial test is therefore not adapted. On the contrary, the permutation test was unaffected by the cross-validation scheme. The influence of the cross-validation was further illustrated on real-data from a brain–computer interface experiment in patients with disorders of consciousness and from an fMRI study on patients with Parkinson disease. Three out of 16 patients with disorders of consciousness had significant accuracy on binomial testing, but only one showed significant accuracy using permutation testing. In the fMRI experiment, the mental imagery of gait could discriminate significantly between idiopathic Parkinson's disease patients and healthy subjects according to the permutation test but not according to the binomial test. Hence, binomial testing could lead to biased estimation of significance and false positive or negative results. In our view, permutation testing is thus recommended for clinical application of classification with cross-validation.

18. Biased binomial assessment of cross-validated estimation of classification accuracies illustrated in diagnosis predictions

Noirhomme, Quentin; Lesenfants, Damien; Gomez, Francisco; Soddu, Andrea; Schrouff, Jessica; Garraux, Gaëtan; Luxen, André; Phillips, Christophe; Laureys, Steven

2014-01-01

Multivariate classification is used in neuroimaging studies to infer brain activation or in medical applications to infer diagnosis. Their results are often assessed through either a binomial or a permutation test. Here, we simulated classification results of generated random data to assess the influence of the cross-validation scheme on the significance of results. Distributions built from classification of random data with cross-validation did not follow the binomial distribution. The binomial test is therefore not adapted. On the contrary, the permutation test was unaffected by the cross-validation scheme. The influence of the cross-validation was further illustrated on real-data from a brain–computer interface experiment in patients with disorders of consciousness and from an fMRI study on patients with Parkinson disease. Three out of 16 patients with disorders of consciousness had significant accuracy on binomial testing, but only one showed significant accuracy using permutation testing. In the fMRI experiment, the mental imagery of gait could discriminate significantly between idiopathic Parkinson's disease patients and healthy subjects according to the permutation test but not according to the binomial test. Hence, binomial testing could lead to biased estimation of significance and false positive or negative results. In our view, permutation testing is thus recommended for clinical application of classification with cross-validation. PMID:24936420

19. Bayesian Estimation Based on Rayleigh Progressive Type II Censored Data with Binomial Removals

Reza Azimi

2013-01-01

the number of units removed at each failure time has a binomial distribution. We use the maximum likelihood and Bayesian procedures to obtain the estimators of parameter and reliability function of Rayleigh distribution. We also construct the confidence intervals for the parameter of Rayleigh distribution. Monte Carlo simulation method is used to generate a progressive type II censored data with binomial removals from Rayleigh distribution, and then these data are used to compute the point and interval estimations of the parameter and compare both the methods used with different random schemes.

20. On The Effectiveness Of The Negative Binomial Approximation In A Multi-Echelon Inventory Model: A Mathematical Analysis

Solis, Adriano O.; Schmidt, Charles P.; Conerly, Michael D.

2007-09-01

Graves (1996) developed a multi-echelon inventory model and used a negative binomial distribution to approximate the distribution of a random variable in the model. Two earlier multi-echelon inventory studies (Graves, 1985; Lee and Moinzadeh, 1987) have similarly used negative binomial approximations. Only computational evidence has been offered in support of the approximations. We provide, for the latest model (Graves, 1996), a mathematical analysis of the effectiveness of such an approximation.

1. The Binomial - Medicine/Art [In Bulgarian

E. Pavlov

2008-12-01

Full Text Available The article considers the binomial – medicine/art. Medical treatment of body should be combined with the medical treatment of soul. ‘A sound mind in a sound body’ can be achieved when art supports medicine. Semiotic aspects of medicine are discussed. The introduction of such ideas to medical education is recommended.

2. Large Deviation Results for Generalized Compound Negative Binomial Risk Models

Fan-chao Kong; Chen Shen

2009-01-01

In this paper we extend and improve some results of the large deviation for random sums of random variables.Let {Xn;n≥1} be a sequence of non-negative,independent and identically distributed random variables with common heavy-tailed distribution function F and finite mean μ∈R+,{N(n);n≥0} be a sequence of negative binomial distributed random variables with a parameter p ∈(0,1),n≥0,let {M(n);n≥0} be a Poisson process with intensity λ0.Suppose {N(n);n≥0},{Xn;n≥1} and {M(n);n≥0} are mutually results.These results can be applied to certain problems in insurance and finance.

3. Pooling overdispersed binomial data to estimate event rate

Chan K Arnold

2008-08-01

Full Text Available Abstract Background The beta-binomial model is one of the methods that can be used to validly combine event rates from overdispersed binomial data. Our objective is to provide a full description of this method and to update and broaden its applications in clinical and public health research. Methods We describe the statistical theories behind the beta-binomial model and the associated estimation methods. We supply information about statistical software that can provide beta-binomial estimations. Using a published example, we illustrate the application of the beta-binomial model when pooling overdispersed binomial data. Results In an example regarding the safety of oral antifungal treatments, we had 41 treatment arms with event rates varying from 0% to 13.89%. Using the beta-binomial model, we obtained a summary event rate of 3.44% with a standard error of 0.59%. The parameters of the beta-binomial model took the values of 1.24 for alpha and 34.73 for beta. Conclusion The beta-binomial model can provide a robust estimate for the summary event rate by pooling overdispersed binomial data from different studies. The explanation of the method and the demonstration of its applications should help researchers incorporate the beta-binomial method as they aggregate probabilities of events from heterogeneous studies.

4. Maximum Likelihood Estimation of the Negative Binomial Dispersion Parameter for Highly Overdispersed Data, with Applications to Infectious Diseases

James O Lloyd-Smith

2007-01-01

Background. The negative binomial distribution is used commonly throughout biology as a model for overdispersed count data, with attention focused on the negative binomial dispersion parameter, κ. A substantial literature exists on the estimation of κ, but most attention has focused on datasets that are not highly overdispersed (i.e., those with κ≥1), and the accuracy of confidence intervals estimated for κ is typically not explored. Methodology. This article presents a simulation study explo...

5. Lognormal and Gamma Mixed Negative Binomial Regression

Zhou, Mingyuan; Li, Lingbo; Dunson, David; Carin, Lawrence

2012-01-01

In regression analysis of counts, a lack of simple and efficient algorithms for posterior computation has made Bayesian approaches appear unattractive and thus underdeveloped. We propose a lognormal and gamma mixed negative binomial (NB) regression model for counts, and present efficient closed-form Bayesian inference; unlike conventional Poisson models, the proposed approach has two free parameters to include two different kinds of random effects, and allows the incorporation of prior inform...

6. Binomial menu auctions in government formation

Breitmoser, Yves

2011-01-01

In a menu auction, players submit bids for all choices the auctioneer A can make, and A then makes the choice that maximizes the sum of bids. In a binomial menu auction (BMA), players submit acceptance sets (indicating which choices they would support), and A chooses the option that maximizes his utility subject to acceptance of the respective players. Monetary transfers may be implicit, but players may also bid by offering "favors" and the like. BMAs provide a unified representation of both ...

7. Fractional differences and sums with binomial coefficients

2012-01-01

In fractional calculus there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Gr\\"{u}nwald-Litnikov fractional derivatives. In this article we formulate the delta and nabla discrete versions f...

8. Multiplicity dependent and non-binomial efficiency corrections for particle number cumulants

2016-01-01

In this note we extend previous work on efficiency corrections for cumulant measurements [1,2]. We will discuss the limitations of the methods presented in these papers. Specifically we will consider multiplicity dependent efficiencies as well as a non-binomial efficiency distributions. We will discuss the most simple and straightforward methods to implement those corrections.

9. Binomial moment equations for stochastic reaction systems.

Barzel, Baruch; Biham, Ofer

2011-04-15

A highly efficient formulation of moment equations for stochastic reaction networks is introduced. It is based on a set of binomial moments that capture the combinatorics of the reaction processes. The resulting set of equations can be easily truncated to include moments up to any desired order. The number of equations is dramatically reduced compared to the master equation. This formulation enables the simulation of complex reaction networks, involving a large number of reactive species much beyond the feasibility limit of any existing method. It provides an equation-based paradigm to the analysis of stochastic networks, complementing the commonly used Monte Carlo simulations. PMID:21568538

10. A Note on Teaching Binomial Confidence Intervals

Santner, Thomas J.

1997-01-01

For constructing confidence intervals for a binomial proportion $p$, Simon (1996, Teaching Statistics) advocates teaching one of two large-sample alternatives to the usual $z$-intervals $\\hat{p} \\pm 1.96 \\times S.E(\\hat{p})$ where $S.E.(\\hat{p}) = \\sqrt{ \\hat{p} \\times (1 - \\hat{p})/n}$. His recommendation is based on the comparison of the closeness of the achieved coverage of each system of intervals to their nominal level. This teaching note shows that a different alternative to $z$-interv...

11. A Mechanistic Beta-Binomial Probability Model for mRNA Sequencing Data.

Smith, Gregory R; Birtwistle, Marc R

2016-01-01

A main application for mRNA sequencing (mRNAseq) is determining lists of differentially-expressed genes (DEGs) between two or more conditions. Several software packages exist to produce DEGs from mRNAseq data, but they typically yield different DEGs, sometimes markedly so. The underlying probability model used to describe mRNAseq data is central to deriving DEGs, and not surprisingly most softwares use different models and assumptions to analyze mRNAseq data. Here, we propose a mechanistic justification to model mRNAseq as a binomial process, with data from technical replicates given by a binomial distribution, and data from biological replicates well-described by a beta-binomial distribution. We demonstrate good agreement of this model with two large datasets. We show that an emergent feature of the beta-binomial distribution, given parameter regimes typical for mRNAseq experiments, is the well-known quadratic polynomial scaling of variance with the mean. The so-called dispersion parameter controls this scaling, and our analysis suggests that the dispersion parameter is a continually decreasing function of the mean, as opposed to current approaches that impose an asymptotic value to the dispersion parameter at moderate mean read counts. We show how this leads to current approaches overestimating variance for moderately to highly expressed genes, which inflates false negative rates. Describing mRNAseq data with a beta-binomial distribution thus may be preferred since its parameters are relatable to the mechanistic underpinnings of the technique and may improve the consistency of DEG analysis across softwares, particularly for moderately to highly expressed genes. PMID:27326762

12. A Mechanistic Beta-Binomial Probability Model for mRNA Sequencing Data.

Gregory R Smith

Full Text Available A main application for mRNA sequencing (mRNAseq is determining lists of differentially-expressed genes (DEGs between two or more conditions. Several software packages exist to produce DEGs from mRNAseq data, but they typically yield different DEGs, sometimes markedly so. The underlying probability model used to describe mRNAseq data is central to deriving DEGs, and not surprisingly most softwares use different models and assumptions to analyze mRNAseq data. Here, we propose a mechanistic justification to model mRNAseq as a binomial process, with data from technical replicates given by a binomial distribution, and data from biological replicates well-described by a beta-binomial distribution. We demonstrate good agreement of this model with two large datasets. We show that an emergent feature of the beta-binomial distribution, given parameter regimes typical for mRNAseq experiments, is the well-known quadratic polynomial scaling of variance with the mean. The so-called dispersion parameter controls this scaling, and our analysis suggests that the dispersion parameter is a continually decreasing function of the mean, as opposed to current approaches that impose an asymptotic value to the dispersion parameter at moderate mean read counts. We show how this leads to current approaches overestimating variance for moderately to highly expressed genes, which inflates false negative rates. Describing mRNAseq data with a beta-binomial distribution thus may be preferred since its parameters are relatable to the mechanistic underpinnings of the technique and may improve the consistency of DEG analysis across softwares, particularly for moderately to highly expressed genes.

13. NBLDA: Negative Binomial Linear Discriminant Analysis for RNA-Seq Data

Dong, Kai; Zhao, Hongyu; Wan, Xiang; Tong, Tiejun

2015-01-01

RNA-sequencing (RNA-Seq) has become a powerful technology to characterize gene expression profiles because it is more accurate and comprehensive than microarrays. Although statistical methods that have been developed for microarray data can be applied to RNA-Seq data, they are not ideal due to the discrete nature of RNA-Seq data. The Poisson distribution and negative binomial distribution are commonly used to model count data. Recently, Witten (2011) proposed a Poisson linear discriminant ana...

14. Marginalized zero-inflated negative binomial regression with application to dental caries.

Preisser, John S; Das, Kalyan; Long, D Leann; Divaris, Kimon

2016-05-10

The zero-inflated negative binomial regression model (ZINB) is often employed in diverse fields such as dentistry, health care utilization, highway safety, and medicine to examine relationships between exposures of interest and overdispersed count outcomes exhibiting many zeros. The regression coefficients of ZINB have latent class interpretations for a susceptible subpopulation at risk for the disease/condition under study with counts generated from a negative binomial distribution and for a non-susceptible subpopulation that provides only zero counts. The ZINB parameters, however, are not well-suited for estimating overall exposure effects, specifically, in quantifying the effect of an explanatory variable in the overall mixture population. In this paper, a marginalized zero-inflated negative binomial regression (MZINB) model for independent responses is proposed to model the population marginal mean count directly, providing straightforward inference for overall exposure effects based on maximum likelihood estimation. Through simulation studies, the finite sample performance of MZINB is compared with marginalized zero-inflated Poisson, Poisson, and negative binomial regression. The MZINB model is applied in the evaluation of a school-based fluoride mouthrinse program on dental caries in 677 children. PMID:26568034

15. Improving Convergence of Binomial Schemes and the Edgeworth Expansion

Alona Bock

2016-05-01

Full Text Available Binomial trees are very popular in both theory and applications of option pricing. As they often suffer from an irregular convergence behavior, improving this is an important task. We build upon a new version of the Edgeworth expansion for lattice models to construct new and quickly converging binomial schemes with a particular application to barrier options.

16. Penggunaan Model Binomial Pada Penentuan Harga Opsi Saham Karyawan

Dara Puspita Anggraeni

2015-11-01

Full Text Available Binomial Model for Valuing Employee Stock Options. Employee Stock Options (ESO differ from standard exchange-traded options. The three main differences in a valuation model for employee stock options : Vesting Period, Exit Rate and Non-Transferability. In this thesis, the model for valuing employee stock options discussed. This model are implement with a generalized binomial model.

17. Fractional Sums and Differences with Binomial Coefficients

2013-01-01

Full Text Available In fractional calculus, there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Grünwald-Letnikov fractional derivatives. In this paper we formulate the delta and nabla discrete versions for left and right fractional integrals and derivatives representing the second approach. Then, we use the discrete version of the Q-operator and some discrete fractional dual identities to prove that the presented fractional differences and sums coincide with the discrete Riemann ones describing the first approach.

18. Growth Estimators and Confidence Intervals for the Mean of Negative Binomial Random Variables with Unknown Dispersion

David Shilane; Derek Bean

2013-01-01

The negative binomial distribution becomes highly skewed under extreme dispersion. Even at moderately large sample sizes, the sample mean exhibits a heavy right tail. The standard normal approximation often does not provide adequate inferences about the data's expected value in this setting. In previous work, we have examined alternative methods of generating confidence intervals for the expected value. These methods were based upon Gamma and Chi Square approximations or tail probability boun...

19. Beta-Negative Binomial Process and Exchangeable Random Partitions for Mixed-Membership Modeling

Zhou, Mingyuan

2014-01-01

The beta-negative binomial process (BNBP), an integer-valued stochastic process, is employed to partition a count vector into a latent random count matrix. As the marginal probability distribution of the BNBP that governs the exchangeable random partitions of grouped data has not yet been developed, current inference for the BNBP has to truncate the number of atoms of the beta process. This paper introduces an exchangeable partition probability function to explicitly describe how the BNBP clu...

20. Emergence of Independent Candidates:A Negative Binomial Regression Model of anIndian Parliamentary Election

Bhattacharya, Kaushik

2010-01-01

The paper specifies a model of the first-past-the-post (FPTP) electoral system in which political parties themselves float independent candidates to gain electoral advantage, leading to a Prisoners’ Dilemma type game where each party tries to out-maneuver one another. Imposing some intuitively appealing assumptions on this game, we show that the total number of independent candidates across constituencies would follow a Negative Binomial distribution. Empirical results for the 2004 parliament...

1. Bayesian Estimation of Negative Binomial Parameters with Applications to RNA-Seq Data

Leon-Novelo, Luis; Claudio FUENTES; Emerson, Sarah

2015-01-01

RNA-Seq data characteristically exhibits large variances, which need to be appropriately accounted for in the model. We first explore the effects of this variability on the maximum likelihood estimator (MLE) of the overdispersion parameter of the negative binomial distribution, and propose instead the use an estimator obtained via maximization of the marginal likelihood in a conjugate Bayesian framework. We show, via simulation studies, that the marginal MLE can better control this variation ...

2. The destructive negative binomial cure rate model with a latent activation scheme

2013-01-01

A new flexible cure rate survival model is developed where the initial number of competing causes of the event of interest (say lesions or altered cells) follow a compound negative binomial (NB) distribution. This model provides a realistic interpretation of the biological mechanism of the event of interest as it models a destructive process of the initial competing risk factors and records only the damaged portion of the original number of risk factors. Besides, it also acc...

3. Empat Model Aproksimasi Binomial Harga Saham Model black-Scholes

Abdul Aziz

2009-11-01

Full Text Available Kami akan menyajikan empat bentuk nilai parameter-parameter u, d, dan p dalam model Binomial harga saham, yang dihasilkan dengan menggunakan penyamaan ekspektasi dan variansi model diskrit dengan kontinu. Metode pertama menggunakan asumsi u . d = 1, yang mana metode ini dapat menghasilkan tiga bentuk solusi untuk parameter-parameter u, d, dan p dalam model Binomial harga saham. Metode kedua menggunakan asumsi p = 0,5. Dari kedua metode ini ternyata dapat dihasilkan empat bentuk solusi u, d, dan p yang berbeda dan akan dibandingkan hasilnya dalam pendekatan nilai option dalam model Binomial dengan model Black-Scholes.

4. Parameter Estimation Based on the Frèchet Progressive Type II Censored Data with Binomial Removals

Mohamed Mubarak

2012-01-01

Full Text Available This paper considers the estimation problem for the Frèchet distribution under progressive Type II censoring with random removals, where the number of units removed at each failure time has a binomial distribution. We use the maximum likelihood method to obtain the estimators of parameters and derive the sampling distributions of the estimators, and we also construct the confidence intervals for the parameters and percentile of the failure time distribution.

5. Fuzzy Binomial Option pricing model: a comparison of the fuzzy binomials and modified numerical methods

Kanthamanond, Piti

2009-01-01

The fuzzy set concept can be applied to the derivative pricing model to cover the uncertainty in the market. The aim of this study is to modify existing fuzzy binomial option pricing models and tests them with several lattice numerical methods to compare the performance and clarify the benefit and necessity of the fuzzy models over the non-fuzzy models. The empirical validation of these models on the artificial data and on the S&P 100 index options is provided with the various tests which cov...

6. Application of binomial-edited CPMG to shale characterization

Washburn, Kathryn E.; Birdwell, Justin E.

2014-01-01

Unconventional shale resources may contain a significant amount of hydrogen in organic solids such as kerogen, but it is not possible to directly detect these solids with many NMR systems. Binomial-edited pulse sequences capitalize on magnetization transfer between solids, semi-solids, and liquids to provide an indirect method of detecting solid organic materials in shales. When the organic solids can be directly measured, binomial-editing helps distinguish between different phases. We applied a binomial-edited CPMG pulse sequence to a range of natural and experimentally-altered shale samples. The most substantial signal loss is seen in shales rich in organic solids while fluids associated with inorganic pores seem essentially unaffected. This suggests that binomial-editing is a potential method for determining fluid locations, solid organic content, and kerogen–bitumen discrimination.

7. Randomized Binomial Tree and Pricing of American-Style Options

Hu Xiaoping

2014-01-01

Full Text Available Randomized binomial tree and methods for pricing American options were studied. Firstly, both the completeness and the no-arbitrage conditions in the randomized binomial tree market were proved. Secondly, the description of the node was given, and the cubic polynomial relationship between the number of nodes and the time steps was also obtained. Then, the characteristics of paths and storage structure of the randomized binomial tree were depicted. Then, the procedure and method for pricing American-style options were given in a random binomial tree market. Finally, a numerical example pricing the American option was illustrated, and the sensitivity analysis of parameter was carried out. The results show that the impact of the occurrence probability of the random binomial tree environment on American option prices is very significant. With the traditional complete market characteristics of random binary and a stronger ability to describe, at the same time, maintaining a computational feasibility, randomized binomial tree is a kind of promising method for pricing financial derivatives.

8. Una prueba de razón de verosimilitudes para discriminar entre la distribución Poisson, Binomial y Binomial Negativa.

López Martínez, Laura Elizabeth

2010-01-01

En este trabajo se realiza inferencia estadística en la distribución Binomial Negativa Generalizada (BNG) y los modelos que anida, los cuales son Binomial, Binomial Negativa y Poisson. Se aborda el problema de estimación de parámetros en la distribución BNG y se propone una prueba de razón de verosimilitud generalizada para discernir si un conjunto de datos se ajusta en particular al modelo Binomial, Binomial Negativa o Poisson. Además, se estudian las potencias y tamaños de la prueba p...

9. Renormdynamics, multiparticle production, negative binomial distribution, and Riemann zeta function

Makhaldiani, N. V.

2013-09-01

After short introduction, we consider different aspects of the renormdynamics. Then scaling functions of the multiparticle production processes and corresponding stochastic dynamics are considered. Nonperturbative quasi-particle dynamics is considered on the base of the toy QCD- O( N)-sigma model. Last section concerns to the NBD-Riemann zeta function connection.

10. Renormdynamics, multiparticle production, negative binomial distribution, and Riemann zeta function

Makhaldiani, N. V., E-mail: mnv@jinr.ru [Joint Institute for Nuclear Research (Russian Federation)

2013-09-15

After short introduction, we consider different aspects of the renormdynamics. Then scaling functions of the multiparticle production processes and corresponding stochastic dynamics are considered. Nonperturbative quasi-particle dynamics is considered on the base of the toy QCD-O(N)-sigma model. Last section concerns to the NBD-Riemann zeta function connection.

11. QUANTUM THEORY FOR THE BINOMIAL MODEL IN FINANCE THEORY

CHEN Zeqian

2004-01-01

In this paper, a quantum model for the binomial market in finance is proposed. We show that its risk-neutral world exhibits an intriguing structure as a disk in the unit ball of R3, whose radius is a function of the risk-free interest rate with two thresholds which prevent arbitrage opportunities from this quantum market. Furthermore, from the quantum mechanical point of view we re-deduce the Cox-Ross-Rubinstein binomial option pricing formula by considering Maxwell-Boltzmann statistics of the system of N distinguishable particles.

12. Properties of some test of equality of two negative binomial samples

Hübnerová, Zuzana; Doudová, Lucie

2013-10-01

When comparing two independent samples from the negative binomial distribution the test of equality of their expected values might be of interest. In this case, the assumptions on the nuisance shape parameter determine the form of suitable test statistic. This paper focuses on test based on Wald statistic under an assumption that the shape parameters in the two samples are known and equal. However, the common shape parameter might be replaced by its estimate, either statistical or expert. We discuss the consequences of this procedure on the power as well as on the size of the test (probability of rejection of H0 when it is true).

13. A binomial stochastic kinetic approach to the Michaelis-Menten mechanism

Lente, Gábor

2013-05-01

This Letter presents a new method that gives an analytical approximation of the exact solution of the stochastic Michaelis-Menten mechanism without computationally demanding matrix operations. The method is based on solving the deterministic rate equations and then using the results as guiding variables of calculating probability values using binomial distributions. This principle can be generalized to a number of different kinetic schemes and is expected to be very useful in the evaluation of measurements focusing on the catalytic activity of one or a few individual enzyme molecules.

14. Selecting the highest probability in binomial or multinomial trials

Levin, Bruce; Robbins, Herbert

1981-01-01

Some sequential procedures are considered for selecting the binomial population with largest success probability or for selecting the multinomial outcome with highest cell probability. Procedures with and without sequential elimination of inferior populations are evaluated with respect to the expected probability of the population selected.

15. Iterated binomial sums and their associated iterated intergrals

Ablinger, J.; Schneider, C. [Johannes Kepler Univ., Linz (Austria). Research Inst. for Symbolic Computation; Bluemlein, J.; Raab, C.G. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany)

2014-06-15

We consider finite iterated generalized harmonic sums weighted by the binomial ({sup 2k}{sub k}) in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator insertions starting at 3-loop order in the coupling constant and extends the classes of the nested harmonic, generalized harmonic and cyclotomic sums. The binomially weighted sums are associated by the Mellin transform to iterated integrals over square-root valued alphabets. The values of the sums for N→∞ and the iterated integrals at x=1 lead to new constants, extending the set of special numbers given by the multiple zeta values, the cyclotomic zeta values and special constants which emerge in the limit N→∞ of generalized harmonic sums. We develop algorithms to obtain the Mellin representations of these sums in a systematic way. They are of importance for the derivation of the asymptotic expansion of these sums and their analytic continuation to N element of C. The associated convolution relations are derived for real parameters and can therefore be used in a wider context, as e.g. for multi-scale processes. We also derive algorithms to transform iterated integrals over root-valued alphabets into binomial sums. Using generating functions we study a few aspects of infinite (inverse) binomial sums.

16. Using Binomial Tree Pricing Model in a Fuzzy Market

YOU Su-rong; LU Yun-sheng

2007-01-01

A model of using binomial tree pricing formulae in a fuzzy market is proposed. In the fuzzy market, a price interval can be got according to the belief degree. The rule for the reasonability of the price interval is proposed. The explicit expression of the interval is discussed in some special settings.

17. Iterated binomial sums and their associated iterated intergrals

We consider finite iterated generalized harmonic sums weighted by the binomial (2kk) in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator insertions starting at 3-loop order in the coupling constant and extends the classes of the nested harmonic, generalized harmonic and cyclotomic sums. The binomially weighted sums are associated by the Mellin transform to iterated integrals over square-root valued alphabets. The values of the sums for N→∞ and the iterated integrals at x=1 lead to new constants, extending the set of special numbers given by the multiple zeta values, the cyclotomic zeta values and special constants which emerge in the limit N→∞ of generalized harmonic sums. We develop algorithms to obtain the Mellin representations of these sums in a systematic way. They are of importance for the derivation of the asymptotic expansion of these sums and their analytic continuation to N element of C. The associated convolution relations are derived for real parameters and can therefore be used in a wider context, as e.g. for multi-scale processes. We also derive algorithms to transform iterated integrals over root-valued alphabets into binomial sums. Using generating functions we study a few aspects of infinite (inverse) binomial sums.

18. Computer generation of initial spatial distribution for cell automata

GuangHua Liu; WenJun Zhang

2011-01-01

The algorithm to generate spatial distribution patterns was developed and presented in this study. Three typical spatial distribution patterns, i.e., Poisson distribution, binomial distribution, and negative binomial distribution were included in the algorithm. The Java program was also provided. The algorithm can be used to generate initial distribution in cell automata modeling.

19. Desingularization of binomial varieties in arbitrary characteristic. Part I. A new resolution function and their properties

Blanco, Rocio

2009-01-01

This paper is devoted to give all the technical constructions and definitions that will lead to the construction of an algorithm of resolution of singularities for binomial ideals. We construct a resolution function that will provide a resolution of singularities for binomial ideals, over a field of arbitrary characteristic. For us, a binomial ideal means an ideal generated by binomial equations without any restriction, including monomials and $p$-th powers, where $p$ is the characteristic of...

20. Comparing of the binomial model and the black-scholes model for options pricing

Lazarova, Limonka; Jolevska-Tuneska, Biljana; Atanasova-Pacemska, Tatjana

2014-01-01

In this paper will be considered the simple binomial model with one and more periods. It will be given the correspondence between binomial model and the Black-Scholes model for option pricing and also will be shown that the binomial model is more simple then the continuous Black- Scholes model from pedagogical point of view.

1. EXPECTED PRESENT VALUE OF TOTAL DIVIDENDS IN THE COMPOUND BINOMIAL MODEL WITH DELAYED CLAIMS AND RANDOM INCOME

周杰明; 莫晓云; 欧辉; 杨向群

2013-01-01

In this paper, a compound binomial model with a constant dividend barrier and random income is considered. Two types of individual claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim and may be delayed for one time period with a certain probability. The premium income is assumed to another binomial process to capture the uncertainty of the customer’s arrivals and payments. A system of difference equations with certain boundary conditions for the expected present value of total dividend payments prior to ruin is derived and solved. Explicit results are obtained when the claim sizes are Kn distributed or the claim size distributions have finite support. Numerical results are also provided to illustrate the impact of the delay of by-claims on the expected present value of dividends.

2. Simulation-Based Estimation of the Structural Errors-in-Variables Negative Binomial Regression Model with an Application

Jie Q. Guo; Tong Li

2001-01-01

This paper studies the effects and estimation of errors-in-variables negative binomial regression model. We prove that in the presence of measurement errors, in general, maximum likelihood estimator of the overdispersion using the observed data is biased upward. We adopt a structural approach assuming that the distribution of the latent variables is known and propose a simulation-based corrected maximum likelihood estimator and a simulation-based corrected score estimator to estimate the erro...

3. An Alternative Formation Theory of Beat. (II) Revelations of Recursion Formulas of the Reflected X-rays and the Anomalous Transmission and Absorption by the Binomial Theorem

Nakajima, Tetsuo

2008-11-01

The recursion formulas for the photon paths in the Borrmann triangle, which satisfy a new modified Pascal triangle can be derived from the binomial theorem by regarding the permutation of the stochastic variables of the diffracted and transmitted X-ray photons. The Borrmann triangle for the n-multiple X-ray reflections expanded by the n-degree binomial distribution consists of the two sub-triangles given by the ( n-1)-degree binomial distribution of the diffracted and transmitted photons. The former sub-triangle shows perfectly flawless symmetry but the latter one shows inevitable asymmetry. A reasonable understanding of both the high intense and very weak photon flows in the Borrmann triangle, which are popularly known as the anomalous transmission and absorption, respectively, are derived from the binomial theorem. Incident photons irradiated at a point O that forms the vertex of the Borrmann triangle propagate through the bypasses parallel to only the complementary half of the integral whole median with the high probabilities from the binomial theorem and emanate them from a short width slit of overline{O'O''} on the base of the high intense photon flow Borrmann triangle ▵ OO' O″, which can be defined by the standard deviation of the normal distribution. The parallel paths to the whole median also pass the very weak photon flows from the high power exponent of d multinomials through the triangle ▵ OO' O″. Both the above contrastive photon flows could coexist in ▵ OO' O″ based upon the complementary rivalry duality from the binomial theorem of ( d+ t) n =1, including the very weak photon flows from the high power exponent of t multinomials near both sides of the Borrmann triangle.

4. Binomial Approximations for Barrier Options of Israeli Style

Dolinsky, Yan

2009-01-01

We show that prices and shortfall risks of game (Israeli) barrier options in a sequence of binomial approximations of the Black--Scholes (BS) market converge to the corresponding quantities for similar game barrier options in the BS market with path dependent payoffs and the speed of convergence is estimated, as well. The results are new also for usual American style options and they are interesting from the computational point of view, as well, since in binomial markets these quantities can be obtained via dynamical programming algorithms. The paper continues the study of [11]and [7] but requires substantial additional arguments in view of pecularities of barrier options which, in particular, destroy the regularity of payoffs needed in the above papers.

5. On pricing futures options on random binomial tree

The discrete-time approach to real option valuation has typically been implemented in the finance literature using a binomial tree framework. Instead we develop a new model by randomizing the environment and call such model a random binomial tree. Whereas the usual model has only one environment (u, d) where the price of underlying asset can move by u times up and d times down, and pair (u, d) is constant over the life of the underlying asset, in our new model the underlying security is moving in two environments namely (u1, d1) and (u2, d2). Thus we obtain two volatilities σ1 and σ2. This new approach enables calculations reflecting the real market since it consider the two states of market normal and extra ordinal. In this paper we define and study Futures options for such models.

6. Central Binomial Sums, Multiple Clausen Values and Zeta Values

Borwein, J M; Kamnitzer, J

2000-01-01

We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Ap\\'ery sums). The study of non-alternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experimental results involving polylogarithms in the golden ratio. In the non-alternating case, there is a strong connection to polylogarithms of the sixth root of unity, encountered in the 3-loop Feynman diagrams of {\\tt hep-th/9803091} and subsequently in hep-ph/9910223, hep-ph/9910224, cond-mat/9911452 and hep-th/0004010.

7. Determination of finite-difference weights using scaled binomial windows

Chu, Chunlei

2012-05-01

The finite-difference method evaluates a derivative through a weighted summation of function values from neighboring grid nodes. Conventional finite-difference weights can be calculated either from Taylor series expansions or by Lagrange interpolation polynomials. The finite-difference method can be interpreted as a truncated convolutional counterpart of the pseudospectral method in the space domain. For this reason, we also can derive finite-difference operators by truncating the convolution series of the pseudospectral method. Various truncation windows can be employed for this purpose and they result in finite-difference operators with different dispersion properties. We found that there exists two families of scaled binomial windows that can be used to derive conventional finite-difference operators analytically. With a minor change, these scaled binomial windows can also be used to derive optimized finite-difference operators with enhanced dispersion properties. © 2012 Society of Exploration Geophysicists.

8. Generalized binomial transform applied to the divergent series

2015-01-01

The divergent series for a function defined through Lapalce integral and the ground state energy of the quartic anharmonic oscillator to large orders are studied to test the generalized binomial transform which is the renamed version of $\\delta$-expansion proposed recently. We show that, by the use of the generalized binomial transform, the values of functions in the limit of zero of an argument is approximately computable from the series expansion around the infinity of the same argument. In the Laplace integral, we investigate the subject in detail with the aid of Mellin transform. In the anharmonic oscillator, we compute the strong coupling limit of the ground state energy and also the expansion coefficients at strong coupling from the weak coupling perturbation series. The obtained result is compared with that of the linear delta expansion.

9. Comparing the Convergence Behaviour of Binomial and Trinomial Models

Mehmet Horasanli

2007-01-01

An American option differs from a European one by the early exercise possibility. An American option can be exercised at any time up to the maturity date. In general, there is unfortunately no analytical solution to the American option problem. Binomial and trinomial approximations are useful to solve this problem but using a lattice model introduces approximation error. Both models have the property of convergence to Black & Scholes prices thus; can be used alternatively to solve the Black &...

10. Comparing the Convergence Behaviour of Binomial and Trinomial Models

Mehmet Horasanli

2008-01-01

An American option differs from a European one by the early exercise possibility. An American option can be exercised at any time up to the maturity date. In general, there is unfortunately no analytical solution to the American option problem. Binomial and trinomial approximations are useful to solve this problem but using a lattice model introduces approximation error. Both models have the property of convergence to Black & Scholes prices thus; can be used alternatively to solve the Black &...

11. On the least common multiple of $q$-binomial coefficients

Guo, Victor J. W.

2009-01-01

In this paper, we prove the following identity $$\\lcm({n\\brack 0}_q,{n\\brack 1}_q,...,{n\\brack n}_q) =\\frac{\\lcm([1]_q,[2]_q,...,[n+1]_q)}{[n+1]_q},$$ where ${n\\brack k}_q$ denotes the $q$-binomial coefficient and $[n]_q=\\frac{1-q^n}{1-q}$. This result is a $q$-analogue of an identity of Farhi [Amer. Math. Monthly, November (2009)].

12. The combinatorial structure of beta negative binomial processes

Heaukulani, Creighton; Roy, Daniel M.

2013-01-01

We characterize the combinatorial structure of conditionally-i.i.d. sequences of negative binomial processes with a common beta process base measure. In Bayesian nonparametric applications, such processes have served as models for latent multisets of features underlying data. Analogously, random subsets arise from conditionally-i.i.d. sequences of Bernoulli processes with a common beta process base measure, in which case the combinatorial structure is described by the Indian buffet process. O...

13. Hits per trial: Basic analysis of binomial data

This report presents simple statistical methods for analyzing binomial data, such as the number of failures in some number of demands. It gives point estimates, confidence intervals, and Bayesian intervals for the failure probability. It shows how to compare subsets of the data, both graphically and by statistical tests, and how to look for trends in time. It presents a compound model when the failure probability varies randomly. Examples and SAS programs are given

14. Beta-Negative Binomial Process and Poisson Factor Analysis

Zhou, Mingyuan; Hannah, Lauren; Dunson, David; Carin, Lawrence

2011-01-01

A beta-negative binomial (BNB) process is proposed, leading to a beta-gamma-Poisson process, which may be viewed as a "multi-scoop" generalization of the beta-Bernoulli process. The BNB process is augmented into a beta-gamma-gamma-Poisson hierarchical structure, and applied as a nonparametric Bayesian prior for an infinite Poisson factor analysis model. A finite approximation for the beta process Levy random measure is constructed for convenient implementation. Efficient MCMC computations are...

15. Influences on leadership behaviour: a binomial logit model

Titus Oshagbemi; Samuel A. Ocholi

2013-01-01

16. Constructions for a bivariate beta distribution

Olkin, Ingram; Trikalinos, Thomas A.

2014-01-01

The beta distribution is a basic distribution serving several purposes. It is used to model data, and also, as a more flexible version of the uniform distribution, it serves as a prior distribution for a binomial probability. The bivariate beta distribution plays a similar role for two probabilities that have a bivariate binomial distribution. We provide a new multivariate distribution with beta marginal distributions, positive probability over the unit square, and correlations over the full ...

17. Two-part zero-inflated negative binomial regression model for quantitative trait loci mapping with count trait.

Moghimbeigi, Abbas

2015-05-01

Poisson regression models provide a standard framework for quantitative trait locus (QTL) mapping of count traits. In practice, however, count traits are often over-dispersed relative to the Poisson distribution. In these situations, the zero-inflated Poisson (ZIP), zero-inflated generalized Poisson (ZIGP) and zero-inflated negative binomial (ZINB) regression may be useful for QTL mapping of count traits. Added genetic variables to the negative binomial part equation, may also affect extra zero data. In this study, to overcome these challenges, I apply two-part ZINB model. The EM algorithm with Newton-Raphson method in the M-step uses for estimating parameters. An application of the two-part ZINB model for QTL mapping is considered to detect associations between the formation of gallstone and the genotype of markers. PMID:25728790

18. Maximum likelihood estimation of the negative binomial dispersion parameter for highly overdispersed data, with applications to infectious diseases.

James O Lloyd-Smith

Full Text Available BACKGROUND: The negative binomial distribution is used commonly throughout biology as a model for overdispersed count data, with attention focused on the negative binomial dispersion parameter, k. A substantial literature exists on the estimation of k, but most attention has focused on datasets that are not highly overdispersed (i.e., those with k>or=1, and the accuracy of confidence intervals estimated for k is typically not explored. METHODOLOGY: This article presents a simulation study exploring the bias, precision, and confidence interval coverage of maximum-likelihood estimates of k from highly overdispersed distributions. In addition to exploring small-sample bias on negative binomial estimates, the study addresses estimation from datasets influenced by two types of event under-counting, and from disease transmission data subject to selection bias for successful outbreaks. CONCLUSIONS: Results show that maximum likelihood estimates of k can be biased upward by small sample size or under-reporting of zero-class events, but are not biased downward by any of the factors considered. Confidence intervals estimated from the asymptotic sampling variance tend to exhibit coverage below the nominal level, with overestimates of k comprising the great majority of coverage errors. Estimation from outbreak datasets does not increase the bias of k estimates, but can add significant upward bias to estimates of the mean. Because k varies inversely with the degree of overdispersion, these findings show that overestimation of the degree of overdispersion is very rare for these datasets.

19. Growth Estimators and Confidence Intervals for the Mean of Negative Binomial Random Variables with Unknown Dispersion

David Shilane

2013-01-01

Full Text Available The negative binomial distribution becomes highly skewed under extreme dispersion. Even at moderately large sample sizes, the sample mean exhibits a heavy right tail. The standard normal approximation often does not provide adequate inferences about the data's expected value in this setting. In previous work, we have examined alternative methods of generating confidence intervals for the expected value. These methods were based upon Gamma and Chi Square approximations or tail probability bounds such as Bernstein's inequality. We now propose growth estimators of the negative binomial mean. Under high dispersion, zero values are likely to be overrepresented in the data. A growth estimator constructs a normal-style confidence interval by effectively removing a small, predetermined number of zeros from the data. We propose growth estimators based upon multiplicative adjustments of the sample mean and direct removal of zeros from the sample. These methods do not require estimating the nuisance dispersion parameter. We will demonstrate that the growth estimators' confidence intervals provide improved coverage over a wide range of parameter values and asymptotically converge to the sample mean. Interestingly, the proposed methods succeed despite adding both bias and variance to the normal approximation.

20. On Some Distributions Arising in Inverse Cluster Sampling

Xekalaki, Evdokia; Panaretos, John

1989-01-01

In this paper, distributions of items sampled inversely in clusters are derived. In particular, negative binomial type of distributions are obtained and their properties are studied. A logarithmic series, type of distribution is also defined as limiting form of the obtained generalized negative binomial distribution

1. Interval Estimation for Binomial Proportion, Poisson Mean, and Negative –binomial Mean

Liu, Luchen

2012-01-01

This paper studies the interval estimation of three discrete distributions: thebinomial distribution, the Poisson distribution and the negative-binomialdistribution. The problem is the chaotic behavior of the coverage probabilityfor the Wald interval. To solve this problem, alternative confidence intervals areintroduced. Coverage probability and expected length are chosen to be thecriteria evaluating the intervals.In this paper, I firstly tested the chaotic behavior of the coverageprobability...

2. Generalized distributions of order k associated with success runs in Bernoulli trials

Gregory A. Tripsiannis; Papathanasiou, Afroditi A.; Philippou, Andreas N.

2003-01-01

In a sequence of independent Bernoulli trials, by counting multidimensional lattice paths in order to compute the probability of a first-passage event, we derive and study a generalized negative binomial distribution of order k, type I, which extends to distributions of order k, the generalized negative binomial distribution of Jain and Consul (1971), and includes as a special case the negative binomial distribution of order k, type I, of Philippou et al. (1983). This new distribution gives r...

3. Estimators for the binomial failure rate common cause model

In Vesely's binomial failure rate model, a system of m components is hit by random shocks which may cause components simultaneously to fail, each component equal probability. Individual components may also fail when no shock has occurred. The data possibilities considered are that caused of single failures are identifiable (as shock or not) or not identifiable. Given data from such a system, non-Bayesian and Bayesian point and interval estimators are found for the various quantities of interest. Residual analyses and hypothesis tests are presented for checking the model assumptions. An example is worked out

4. The binomial edge ideal of a pair of graphs

Ene, Viviana; Hibi, Takayuki; Qureshi, Ayesha Asloob

2012-01-01

We introduce a class of ideals generated by a set of 2-minors of $m\\times n$-matrix of indeterminates indexed by a pair of graphs. This class of ideals is a natural common generalization of binomial edge ideals and ideals generated by adjacent minors. We determine the minimal prime ideals of such ideals and give a lower bound for their degree of nilpotency. In some special cases we compute their Gr\\"obner basis and characterize unmixedness and Cohen--Macaulayness.

5. Reliability Estimation for poisson-exponential model under Progressive type-II censoring data with binomial removal data

Manoj Kumar

2016-03-01

Full Text Available In this paper, a poissoin-exponential distribution(PED is considered as a lifetime model. Its statistical characteristics and important distributional properties are discussed by Louzada-Neto et al.[13]. The method of Maximum likelihood estimation and least square estimation of parameters involved along with reliability and failure rate functions is also studied here. In view of cost and time constraints, Progressive type-II censored data with binomial removals (PT-II CBRs have been used. Finally, a real data example is given to show the practical applications of the paper.

6. THE INTERLINGUAL ERRORS OF ARAB STUDENTS IN THE USE OF ENGLISH BINOMIALS

Mohamed, Abdul minam Mahmod [عبد المنعم محمود محمد

2003-01-01

This study focuses on the interlingual errors made by adult Arab learners in the use of the English binomials. Thus, it adds one more area to those where language transfer has been reported. A total of 73 binomials were collected from the oral and written presentation of the second year university students majoring in English. The incorrect binomials (85%) could all be attributed to negative transfer from Arabic. These were grouped and analyzed under three main categories: grammatical errors,...

7. The Compound Binomial Risk Model with Randomly Charging Premiums and Paying Dividends to Shareholders

Xiong Wang

2013-01-01

Full Text Available Based on characteristics of the nonlife joint-stock insurance company, this paper presents a compound binomial risk model that randomizes the premium income on unit time and sets the threshold for paying dividends to shareholders. In this model, the insurance company obtains the insurance policy in unit time with probability and pays dividends to shareholders with probability when the surplus is no less than . We then derive the recursive formulas of the expected discounted penalty function and the asymptotic estimate for it. And we will derive the recursive formulas and asymptotic estimates for the ruin probability and the distribution function of the deficit at ruin. The numerical examples have been shown to illustrate the accuracy of the asymptotic estimations.

Donoghoe, Mark W; Marschner, Ian C

2016-08-15

Rate differences are an important effect measure in biostatistics and provide an alternative perspective to rate ratios. When the data are event counts observed during an exposure period, adjusted rate differences may be estimated using an identity-link Poisson generalised linear model, also known as additive Poisson regression. A problem with this approach is that the assumption of equality of mean and variance rarely holds in real data, which often show overdispersion. An additive negative binomial model is the natural alternative to account for this; however, standard model-fitting methods are often unable to cope with the constrained parameter space arising from the non-negativity restrictions of the additive model. In this paper, we propose a novel solution to this problem using a variant of the expectation-conditional maximisation-either algorithm. Our method provides a reliable way to fit an additive negative binomial regression model and also permits flexible generalisations using semi-parametric regression functions. We illustrate the method using a placebo-controlled clinical trial of fenofibrate treatment in patients with type II diabetes, where the outcome is the number of laser therapy courses administered to treat diabetic retinopathy. An R package is available that implements the proposed method. Copyright © 2016 John Wiley & Sons, Ltd. PMID:27073156

9. A Generic Multivariate Distribution for Counting Data

Capistrán, Marcos; Christen, J. Andrés

2011-01-01

Motivated by the need, in some Bayesian likelihood free inference problems, of imputing a multivariate counting distribution based on its vector of means and variance-covariance matrix, we define a generic multivariate discrete distribution. Based on blending the Binomial, Poisson and Negative-Binomial distributions, and using a normal multivariate copula, the required distribution is defined. This distribution tends to the Multivariate Normal for large counts and has an approximate pmf versi...

10. Weak convergence to the t-distribution

Christian Schluter; Mark Trede

2011-01-01

We present a new limit theorem for random means: if the sample size is not deterministic but has a negative binomial or geometric distribution, the limit distribution of the normalised random mean is a t-distribution with degrees of freedom depending on the shape parameter of the negative binomial distribution. Thus the limit distribution exhibits exhibits heavy tails, whereas limit laws for random sums do not achieve this unless the summands have in nite variance. The limit law may help expl...

11. Adjusted Wald Confidence Interval for a Difference of Binomial Proportions Based on Paired Data

Bonett, Douglas G.; Price, Robert M.

2012-01-01

Adjusted Wald intervals for binomial proportions in one-sample and two-sample designs have been shown to perform about as well as the best available methods. The adjusted Wald intervals are easy to compute and have been incorporated into introductory statistics courses. An adjusted Wald interval for paired binomial proportions is proposed here and…

12. On sums involving products of three binomial coefficients

Sun, Zhi-Wei

2010-01-01

In this paper we mainly employ the Zeilberger algorithm to study congruences for sums of terms involving products of three binomial coefficients. Let $p>3$ be a prime. We prove that $$\\sum_{k=0}^{p-1}\\frac{\\binom{2k}k^2\\binom{2k}{k+d}}{64^k}\\equiv 0\\pmod{p^2}$$ for all $d\\in\\{0,\\ldots,p-1\\}$ with $d\\equiv (p+1)/2\\pmod2$. If $p\\equiv 1\\pmod4$ and $p=x^2+y^2$ with $x\\equiv 1\\pmod4$ and $y\\equiv 0\\pmod2$, then we show \\sum_{k=0}^{p-1}\\frac{\\binom{2k}k^2\\binom{2k}{k+1}}{(-8)^k}\\equiv 2p-2x^2\\pm...

13. Probabilidades de ruína e o modelo binomial

Jerónimo, Isabel Brandão

2004-01-01

Mestrado em Ciências Actuariais Nesta dissertação, vamos apresentar o modelo binomial composto em tempo discreto e calcular probabilidades de ruína. Do ponto de vista da probabilidade de ruína em tempo finito, em vez de ter em linha de conta o instante temporal em que a ruína ocorre, estudamos a probabilidade da ruína ocorrer na n-ésima indemnizacão e o número de indem¬nizações ocorridas durante o período de recuperação do processo de risco. O nosso objectivo e, não só, obter resultados nu...

14. Calculation Methods for Wallenius’ Noncentral Hypergeometric Distribution

Fog, Agner

2008-01-01

conditional distribution of independent binomial variates given their sum. No reliable calculation method for Wallenius' noncentral hypergeometric distribution has hitherto been described in the literature. Several new methods for calculating probabilities from Wallenius' noncentral hypergeometric...

15. Statistical Properties and Algebraic Characteristics of Quantum Superpositions of Negative Binomial States

WANG XiaoGuang; FU Hong-Chen

2001-01-01

We introduce new kinds of states of quantized radiation fields, which are the superpositions of negative binomial states. They exhibit remarkable nonclassical properties and reduce to Schrodinger cat states in a certain limit.The algebras involved in the even and odd negative binomial states turn out to be generally deformed oscillator algebras.It is found that the even and odd negative binomial states satisfy the same eigenvalue equation with the same eigenvalue and they can be viewed as two-photon nonlinear coherent states. Two methods of generating such the states are proposed.

16. Low reheating temperatures in monomial and binomial inflationary potentials

Rehagen, Thomas

2015-01-01

We investigate the allowed range of reheating temperature values in light of the Planck 2015 results and the recent joint analysis of Cosmic Microwave Background (CMB) data from the BICEP2/Keck Array and Planck experiments, using monomial and binomial inflationary potentials. While the well studied $\\phi^2$ inflationary potential is no longer favored by current CMB data, as well as $\\phi^p$ with $p>2$, a $\\phi^1$ potential and canonical reheating ($w_{re}=0$) provide a good fit to the CMB measurements. In this last case, we find that the Planck 2015 $68\\%$ confidence limit upper bound on the spectral index, $n_s$, implies an upper bound on the reheating temperature of $T_{re}\\lesssim 6\\times 10^{10}\\,{\\rm GeV}$, and excludes instantaneous reheating. The low reheating temperatures allowed by this model open the possiblity that dark matter could be produced during the reheating period instead of when the Universe is radiation dominated, which could lead to very different predictions for the relic density and mo...

17. Microbial comparative pan-genomics using binomial mixture models

Ussery David W

2009-08-01

Full Text Available Abstract Background The size of the core- and pan-genome of bacterial species is a topic of increasing interest due to the growing number of sequenced prokaryote genomes, many from the same species. Attempts to estimate these quantities have been made, using regression methods or mixture models. We extend the latter approach by using statistical ideas developed for capture-recapture problems in ecology and epidemiology. Results We estimate core- and pan-genome sizes for 16 different bacterial species. The results reveal a complex dependency structure for most species, manifested as heterogeneous detection probabilities. Estimated pan-genome sizes range from small (around 2600 gene families in Buchnera aphidicola to large (around 43000 gene families in Escherichia coli. Results for Echerichia coli show that as more data become available, a larger diversity is estimated, indicating an extensive pool of rarely occurring genes in the population. Conclusion Analyzing pan-genomics data with binomial mixture models is a way to handle dependencies between genomes, which we find is always present. A bottleneck in the estimation procedure is the annotation of rarely occurring genes.

18. On Moran's Property of the Poisson Distribution

Panaretos, John

1983-01-01

Two interesting results encountered in the literature concerning the Poisson and the negative binomial distributions are due to MORAN (1952) and PATIL & SESHADRI (1964), respectively. MORAN's result provided a fundamental property of the Poisson distribution. Roughly speaking, he has shown that if Y, Z are independent, non-negative, integer-valued random variables with X=Y | Z then, under some mild restrictions, the conditional distribution of Y | X is binomial if and only if Y, Z are Poi...

19. La aproximación binomial por la normal: Una experiencia de reflexión sobre la práctica Binomial approach for the normal: An Experience of Reflection on Practice

2010-12-01

Full Text Available En este trabajo se describe un proceso de reflexión sobre la enseñanza de la estadística a nivel universitario. Para ello se analiza las dificultades de comprensión de la aproximación de la distribución binomial por la normal, surgida de la práctica docente. Como primera etapa se considera un estudio histórico y el análisis de libros de texto como los elementos que configuran e influyen en el problema profesional. Como consecuencia, se plantea un diseño de implementación mediante representaciones de simulación para facilitar la comprensión progresiva y su generalización al estudio del teorema central del límite.This paper describes a reflection process on the teaching of statistics at university level. It also discusses the difficulties of understanding the binomial approximation of the normal distribution, as a result of the teaching practice. As a first step, a historical study and analysis of textbooks is considered, as well as the elements that shape and influence professional issues. As a result, an implementation design is proposed through simulated representations to facilitate the progressive understanding and generalization to the study of central limit theorem.

20. New Classes of Permutation Binomials and Permutation Trinomials over Finite Fields

Li, Kangquan; Qu, Longjiang; Chen, Xi

2015-01-01

Permutation polynomials over finite fields play important roles in finite fields theory. They also have wide applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, communication theory and so on. Permutation binomials and trinomials attract people's interest due to their simple algebraic form and additional extraordinary properties. In this paper, several new classes of permutation binomials and permutation trinomials are constructed. ...

1. Non-classical properties and algebraic characteristics of negative binomial states in quantized radiation fields

Wang, Xiao-Guang; Pan, Shao-Hua; Yang, Guo-Zhen

1999-01-01

We study the nonclassical properties and algebraic characteristics of the negative binomial states introduced by Barnett recently. The ladder operator formalism and displacement operator formalism of the negative binomial states are found and the algebra involved turns out to be the SU(1,1) Lie algebra via the generalized Holstein-Primarkoff realization. These states are essentially Peremolov's SU(1,1) coherent states. We reveal their connection with the geometric states and find that they ar...

2. On Bivariate Generalized Exponential-Power Series Class of Distributions

Jafari, Ali Akbar; Roozegar, Rasool

2015-01-01

In this paper, we introduce a new class of bivariate distributions by compounding the bivariate generalized exponential and power-series distributions. This new class contains some new sub-models such as the bivariate generalized exponential distribution, the bivariate generalized exponential-poisson, -logarithmic, -binomial and -negative binomial distributions. We derive different properties of the new class of distributions. The EM algorithm is used to determine the maximum likelihood estim...

3. Statistical Yield Modeling for IC Manufacture: Hierarchical Fault Distributions

Bogdanov, Yu. I.; Bogdanova, N. A.; Dshkhunyan, V. L.

2003-01-01

A hierarchical approach to the construction of compound distributions for process-induced faults in IC manufacture is proposed. Within this framework, the negative binomial distribution and the compound binomial distribution are treated as level-1 models. The hierarchical approach to fault distribution offers an integrated picture of how fault density varies from region to region within a wafer, from wafer to wafer within a batch, and so on. A theory of compound-distribution hierarchies is de...

4. Verification of rod safety via confidence intervals for a binomial proportion

The probabilistic safety assessed to a set of N fuel rods assembled in one core of a nuclear power reactor is commonly modelled by Σ i≤N X i, where X 1, ..., X N are independent Bernoulli random variables (rv) with individual probability p i = P (X i = 1) that the ith rod shows no failure during one cycle. This is the probability of the event that the ith rod will not exceed the failure limit during one cycle. The safety standard presently set by the German Reaktor-Sicherheitskommission (Reactor Safety Commission) requires that the expected number of unfailed rods in the core during one cycle is at least N - 1, i.e., E(Σ i≤N X i) = Σ i≤N p i ≥ N - 1, whereby a confidence level of 0.95 for the verification of this condition is demanded. In this paper, we provide an approach, based on the Clopper-Pearson confidence interval for the proportion p of a binomial B(n, p) distribution, how to verify this condition with a confidence level of at least 0.95. We extend our approach to the case, where the set of N fuel rods is arranged in strata, possibly due to different design in each stratum

5. Effects of the Detection Efficiency on Multiplicity Distributions

Tang, A

2013-01-01

In this paper we investigate how a finite detection efficiency affects three popular multiplicity distributions, namely the Poisson, the Binomial and the Negative Binomial distributions. We found that a constant detection efficiency does not change the characteristic of a distribution, while a variable detection efficiency does. We layout a procedure to study the deviation of moments from the baseline distribution due to a variable detection efficiency.

6. Effects of the detection efficiency on multiplicity distributions

Tang, A. H.; Wang, G.

2013-08-01

In this paper we investigate how a finite detection efficiency affects three popular multiplicity distributions, namely, the Poisson, the binomial, and the negative binomial distributions. We found that a multiplicity-independent detection efficiency does not change the characteristic of a distribution, while a multiplicity-dependent detection efficiency does. We layout a procedure to study the deviation of moments and their derivative quantities from the baseline distribution due to a multiplicity-dependent detection efficiency.

7. Effects of the Detection Efficiency on Multiplicity Distributions

Tang, A; Wang, G

2013-01-01

In this paper we investigate how a finite detection efficiency affects three popular multiplicity distributions, namely the Poisson, the Binomial and the Negative Binomial distributions. We found that a multiplicity-independent detection efficiency does not change the characteristic of a distribution, while a multiplicity-dependent detection efficiency does. We layout a procedure to study the deviation of moments and their derivative quantities from the baseline distribution due to a multipli...

8. Disoriented Chiral Condensates, Pion Probability Distributions and Parallels with Disordered System

Mekjian, A. Z.

1999-01-01

A general expression is discussed for pion probability distributions coming from relativistic heavy ion collisions. The general expression contains as limits: 1) The disoriented chiral condensate (DCC), 2) the negative binomial distribution and Pearson type III distribution, 3) a binomial or Gaussian result, 4) and a Poisson distribution. This general expression approximates other distributions such as a signal to noise laser distribution. Similarities and differences of the DCC distribution ...

9. Statistical prop erties of binomial and negative-binomial combinational optical field state and its generation in quantum diffusion channel%二项-负二项组合光场态的光子统计性质及其在量子扩散通道中的生成∗

范洪义; 吴泽

2015-01-01

According to the combinational binomial-negative-binomial distribution, we propose a binomial-negative-binomial combinational optical field state, which can be generated in the process of a Fock state |m⟩⟨m| passing through a quantum-mechanical diffusion channel. We derive the second-order coherence degree formula, g(2) (t)=2− m2+m(m+κt)2 , which is the diffusion constant. We find that in the process of the Fock state undergoing quantum diffusion and becoming classical, the corresponding photon statistics evolves from sub-Poissonian distribution to Poisson distribution and finally goes to a chaotic state. We also find that the more photons in the initial Fock state, the longer time is needed for quantum decoherence.%在组合二项-负二项分布的基础上,提出了二项-负二项组合光场态,这种态能在Fo ck态历经量子扩散通道的过程中实现。导出了此光场的二阶相干度公式, g(2)(t)=2− m2+m(m+κt)2,发现随着时间的推移光场从非经典Fock态变为经典态,光子数m经扩散通道后变成了m+κt,κ是扩散常数,相应的光子统计从亚泊松分布历经泊松分布再变成混沌光；初始Fo ck态的光子数越多,则扩散所需的时间越长。

10. Relacionando las distribuciones binomial negativa y logarítmica vía sus series asociadas

2011-01-01

La distribución binomial negativa está asociada a la serie obtenida de derivar la serie logarítmica. Recíprocamente, la distribución logarítmica está asociada a la serie obtenida de integrar la serie asociada a la distribución binomial negativa. El parámetro del número de fallas de la distribución Binomial negativa es el número de derivadas necesarias para obtener la serie binomial negativa de la serie logarítmica. El razonamiento presentado puede emplearse como un método alternativo para pro...

11. Flood analysis using negative binomial and Generalized Pareto models in partial duration series (PDS)

Bhunya, P. K.; Berndtsson, R.; Jain, Sharad. K.; Kumar, Rakesh

2013-08-01

Two flood analysis estimation schemes, based on, respectively, partial duration series (PDS) and annual maximum series (AMS), are compared. The PDS model assumes a Generalized Pareto (GP) distribution for modeling the flood exceedances above threshold corresponding to a generalized extreme value (GEV) distribution for annual maxima. As a generalization of the common assumption of the Poisson distribution (PD) to count the occurrences of peaks over threshold in the PDS models, the advantage of negative binomial (NB) distribution is explored in this study. The T-year event estimator for the annual maximum distribution corresponding to the parent PDS model is formulated for producing AMS samples consistent with PDS samples which are used in simulations. The performance of the two models in terms of the uncertainty of the T-year event estimator is evaluated in the cases of estimation with the method of probability weighted moments (PWM). In a similar way, the performance of the derived PDS/NB-GP model is compared with the existing PDS/PD-GP model in terms of uncertainty of T-year event estimator using simulation and field data. The results show the T-year event estimate using PDS/NB-GP model yields lower variance compared to PDS/PD-GP models for most cases. However both the models perform similarly at higher return periods more than 300 years, using the ratios of the variance of T-years estimate as an index, and the ratio decreases with an increase in mean number of annual exceedances above threshold (μ). From the results it is observed that both AMS and PDS models yield the same variance when μ varies from 1.4 to 1.65. However, in case of NB distribution the PDS and AMS models gives the same variance of q(T) when variance (σ2) is 1.5 times the mean number of annual exceedance above threshold. The performance of the PDS models and the corresponding AMS models using the available data of Dee (at Cairnton) shows the PDS/NB-GP model to be marginally better at return

12. Entanglement properties between two atoms in the binomial optical field interacting with two entangled atoms

刘堂昆; 张康隆; 陶宇; 单传家; 刘继兵

2016-01-01

The temporal evolution of the degree of entanglement between two atoms in a system of the binomial optical field interacting with two arbitrary entangled atoms is investigated. The influence of the strength of the dipole–dipole interaction between two atoms, probabilities of the Bernoulli trial, and particle number of the binomial optical field on the temporal evolution of the atomic entanglement are discussed. The result shows that the two atoms are always in the entanglement state. Moreover, if and only if the two atoms are initially in the maximally entangled state, the entanglement evolution is not affected by the parameters, and the degree of entanglement is always kept as 1.

13. Entanglement and photon statistics of output fields from beam splitter for binomial state inputs

Zhou Qing-Ping; Fang Mao-Fa

2004-01-01

The entanglement properties are investigated based on linear entropy, and nonclassicalities are examined of output fields from a beam splitter for pure binomial state inputs. It is shown that the properties of the entanglement and the photon statistics of output fields are not only strongly dependent on the parameters of input binomial states but also quite involved with the nature of the beam splitter. The best entanglement can be obtained when the parameters of both input states and the beam splitter are chosen appropriately. Finally, we analyse briefly the distinguishability between the joint input state and the joint output state.

14. On negative binomial approximation to k-runs

Wang, Xiaoxin; Xia, Aihua

2008-01-01

The distributions of the run occurrences for a sequence of independent and identically distributed (i.i.d.) experiments are usually obtained by combinatorial methods (see Balakrishnan and Koutras (2002, Chapter 5)) and the resulting formulae are often very tedious, while the distributions for non i.i.d. experiments are generally intractable. It is therefore of practical interest to find a suitable approximate model with reasonable approximation accuracy. In this paper we ...

15. New Analysis of Cumulant Moments in $e^+e^-$ Collisions by SLD Collaboration by Truncated Multiplicity Distributions

Suzuki, N.; Biyajima, M.; Nakajima, N

1996-01-01

Newly reported normalized cumulant moments of charged particles in $e^+e^-$ collisions by the SLD collaboration are analyzed by the truncated modified negative binomial distribution (MNBD) and the negative binomial distribution (NBD). Calculated result by the MNBD describes the oscillatory behavior of the data much better than that by the NBD.

16. Modelling noise in second generation sequencing forensic genetics STR data using a one-inflated (zero-truncated) negative binomial model

Vilsen, Søren B.; Tvedebrink, Torben; Mogensen, Helle Smidt;

2015-01-01

function and (2) we used the expectation-maximisation, EM, algorithm. The estimated parameters were used to create dynamic, sample specific thresholds for noise removal using marker specific proportions of the negative binomial distribution. Based on data from dilution series experiments (amounts of DNA...... ranging from 100 pg to 2 ng) conducted at The Section of Forensic Genetics, Department of Forensic Medicine, Faculty of Health and Medical Sciences, University of Copenhagen, Denmark, the method was compared to that of a naïve model that implies the removal of reads with a coverage of less than 5–10% of...

17. Overdispersion: Notes on discrete distributions

Bowman, K.O. [Oak Ridge National Lab., TN (United States); Shenton, L.R. [Georgia Univ., Athens, GA (United States); Kastenbaum, M.A. [Ft. Myers, Florida (United States); Broman, K. [Wisconsin Univ., Milwaukee, WI (United States)

1992-09-01

We introduce mixtures of binomial distributions derived by assuming that the probability parameter p varies according to some law. We use the transformation p = exp({minus}t) and consider various appropriate densities for the transformed variables. In the process, the Laplace transform becomes the fundamental entity. Large numbers of new binomial mixtures are generated in this way. Some transformations may involve several variates that lead to multivariate binomial mixtures. An extension of this to the logarithmic distribution, with parameter p, is possible. Frullani integrals and Laplace transforms are encountered. Graphical representations of some of the more significant distributions are given. These include probability functions, regions of validity, and three dimensional representations of probability functions showing the response to variation of parameters when two parameters are involved.

18. Overdispersion: Notes on discrete distributions

Bowman, K.O. (Oak Ridge National Lab., TN (United States)); Shenton, L.R. (Georgia Univ., Athens, GA (United States)); Kastenbaum, M.A. (Ft. Myers, Florida (United States)); Broman, K. (Wisconsin Univ., Milwaukee, WI (United States))

1992-09-01

We introduce mixtures of binomial distributions derived by assuming that the probability parameter p varies according to some law. We use the transformation p = exp([minus]t) and consider various appropriate densities for the transformed variables. In the process, the Laplace transform becomes the fundamental entity. Large numbers of new binomial mixtures are generated in this way. Some transformations may involve several variates that lead to multivariate'' binomial mixtures. An extension of this to the logarithmic distribution, with parameter p, is possible. Frullani integrals and Laplace transforms are encountered. Graphical representations of some of the more significant distributions are given. These include probability functions, regions of validity, and three dimensional representations of probability functions showing the response to variation of parameters when two parameters are involved.

19. Extended Conway-Maxwell-Poisson distribution and its properties and applications

Chakraborty, Subrata; Imoto, Tomoaki

2015-01-01

A new three parameter natural extension of the Conway-Maxwell-Poisson (COM-Poisson) distribution is proposed. This distribution includes the recently proposed COM-Poisson type negative binomial (COM-NB) distribution [Chakraborty, S. and Ong, S. H. (2014): A COM-type Generalization of the Negative Binomial Distribution, Accepted in Communications in Statistics-Theory and Methods] and the generalized COM-Poisson (GCOMP) distribution [Imoto, T. :(2014) A generalized Conway-Maxwell-Poisson distri...

20. Joint Analysis of Binomial and Continuous Traits with a Recursive Model

Varona, Louis; Sorensen, Daniel

2014-01-01

This work presents a model for the joint analysis of a binomial and a Gaussian trait using a recursive parametrization that leads to a computationally efficient implementation. The model is illustrated in an analysis of mortality and litter size in two breeds of Danish pigs, Landrace and Yorkshire...

1. The Euler Series Transformation and the Binomial Identities of Ljunggren, Munarini and Simons

2009-01-01

It is shown that the curious identity of Simons follows immediately from Euler's series transformation formula and also from an identity due to Ljunggren. The relation of Simons' identity to Legendre's polynomials is also discussed. At the end we use the generalized Euler series transformation to obtain two recent binomial identities of Munarini.

2. On the Residues of Binomial Coefficients and Their Products Modulo Prime Powers

CAI Tian Xin; GRANVILLE Andrew

2002-01-01

In this paper, we show several arithmetic properties on the residues of binomial coefficients and their products modulo prime powers, e.g.,((pq-1 (pq-1)/2)≡ (p-1 (p-1)/2)(q-1 (q-1)/2)(modpq),for any distinct odd primes p and q. Meanwhile, we discuss the connections with the prime recognitions.

3. Generating and Revealing a Quantum Superposition of Electromagnetic Field Binomial States in a Cavity

Franco, R. Lo; Compagno, G; Messina, A.; Napoli, A.

2007-01-01

We introduce the $N$-photon quantum superposition of two orthogonal generalized binomial states of electromagnetic field. We then propose, using resonant atom-cavity interactions, non-conditional schemes to generate and reveal such a quantum superposition for the two-photon case in a single-mode high-$Q$ cavity. We finally discuss the implementation of the proposed schemes.

4. Exact Kolmogorov and total variation distances between some familiar discrete distributions

2006-01-01

Full Text Available We give exact closed-form expressions for the Kolmogorov and the total variation distances between Poisson, binomial, and negative binomial distributions with different parameters. In the Poisson case, such expressions are related with the Lambert W function.

5. Deconvoluting preferences and errors: a model for binomial panel data

Fosgerau, Mogens; Nielsen, Søren Feodor

2010-01-01

In many stated choice experiments researchers observe the random variables Vt, Xt, and Yt = 1{U + δxs22A4Xt + εt distributions of U and εt and also the unknown para...

6. A Negative Binomial Regression Model for Accuracy Tests

Hung, Lai-Fa

2012-01-01

Rasch used a Poisson model to analyze errors and speed in reading tests. An important property of the Poisson distribution is that the mean and variance are equal. However, in social science research, it is very common for the variance to be greater than the mean (i.e., the data are overdispersed). This study embeds the Rasch model within an…

7. Frequency distribution of coliforms in water distribution systems.

Christian, R R; Pipes, W O

1983-01-01

Nine small water distribution systems were sampled intensively to determine the patterns of dispersion of coliforms. The frequency distributions of confirmed coliform counts were compatible with either the negative-binomial or the lognormal distribution. They were not compatible with either the Poisson or Poisson-plus-added zeroes distribution. The implications of the use of the lognormal distributional model were further evaluated because of its previous use in water quality studies. The geo...

8. La loi de Pascal restreinte et ses cas particuliers; Constrained Pascal distribution and its special cases

Louis Laurencelle

2012-01-01

Pascal distribution, Pa(r, /pi), also called Negative binomial distribution, pertains to the trial number n at which the first r successes have been obtained, each trial being the realization of a Bernoulli variable with success probability /pi. We review basic concepts of the Bernoulli process and delve into the constrained Pascal distribution, or Negative binomial distribution of order k, Pa(r, k, /pi), which concerns the trial number n at which r successes have been recorded within the las...

9. A New Extension of the Binomial Error Model for Responses to Items of Varying Difficulty in Educational Testing and Attitude Surveys.

James A Wiley

Full Text Available We put forward a new item response model which is an extension of the binomial error model first introduced by Keats and Lord. Like the binomial error model, the basic latent variable can be interpreted as a probability of responding in a certain way to an arbitrarily specified item. For a set of dichotomous items, this model gives predictions that are similar to other single parameter IRT models (such as the Rasch model but has certain advantages in more complex cases. The first is that in specifying a flexible two-parameter Beta distribution for the latent variable, it is easy to formulate models for randomized experiments in which there is no reason to believe that either the latent variable or its distribution vary over randomly composed experimental groups. Second, the elementary response function is such that extensions to more complex cases (e.g., polychotomous responses, unfolding scales are straightforward. Third, the probability metric of the latent trait allows tractable extensions to cover a wide variety of stochastic response processes.

10. Using a Negative Binomial Regression Model for Early Warning at the Start of a Hand Foot Mouth Disease Epidemic in Dalian, Liaoning Province, China.

Qingyu An

Full Text Available The hand foot and mouth disease (HFMD is a human syndrome caused by intestinal viruses like that coxsackie A virus 16, enterovirus 71 and easily developed into outbreak in kindergarten and school. Scientifically and accurately early detection of the start time of HFMD epidemic is a key principle in planning of control measures and minimizing the impact of HFMD. The objective of this study was to establish a reliable early detection model for start timing of hand foot mouth disease epidemic in Dalian and to evaluate the performance of model by analyzing the sensitivity in detectability.The negative binomial regression model was used to estimate the weekly baseline case number of HFMD and identified the optimal alerting threshold between tested difference threshold values during the epidemic and non-epidemic year. Circular distribution method was used to calculate the gold standard of start timing of HFMD epidemic.From 2009 to 2014, a total of 62022 HFMD cases were reported (36879 males and 25143 females in Dalian, Liaoning Province, China, including 15 fatal cases. The median age of the patients was 3 years. The incidence rate of epidemic year ranged from 137.54 per 100,000 population to 231.44 per 100,000population, the incidence rate of non-epidemic year was lower than 112 per 100,000 population. The negative binomial regression model with AIC value 147.28 was finally selected to construct the baseline level. The threshold value was 100 for the epidemic year and 50 for the non- epidemic year had the highest sensitivity(100% both in retrospective and prospective early warning and the detection time-consuming was 2 weeks before the actual starting of HFMD epidemic.The negative binomial regression model could early warning the start of a HFMD epidemic with good sensitivity and appropriate detection time in Dalian.