A generalization of the beta–binomial distribution

Summary.  The paper describes a distribution generated by the Gaussian hypergeometric function that may be seen as a generalization of the beta–binomial distribution. It is expressed as a generalized beta mixture of a binomial distribution. This new mixing distribution allows the existence of a mode and an antimode, which is very useful for fitting some data sets. Two examples illustrate the greater versatility of the new distribution compared with the beta–binomial distribution.     

On upper bounds for the characteristic values of the covariance matrix for multinomial,dirichlet and multivariate hypergeometric distributions

For the characteristic values T1 of the matrix V:=Diag(p)-pp^T with p=(p1,...,pk), p1≥p2≥...≥pk≥pk+1>0 and p1+p2+...+pk+pk+1=1 the inequalities p1≥τ1≥p2≥τ2≥...≥pk≥τk>0 are given by RONNING (1982). These inequalities give, if p and pk+1 are unknown, the upper bound 1≥T1. However, in this note the bound 1/2≥T1 is derived. V is proportional to the covariance matrix for multinomial, Dirichlet and multivariate hypergeometric distributions. A statistical application for the multinomial distribution is given.     

Confidence interval estimation for negative binomial group distribution

Negative binomial group distribution was proposed in the literature which was motivated by inverse sampling when considering group inspection: products are inspected group by group, and the number of non-conforming items of a group is recorded only until the inspection of the whole group is finished. The non-conforming probability p of the population is thus the parameter of interest. In this paper, the confidence interval construction for this parameter is investigated. The common normal approximation and exact method are applied. To overcome the drawbacks of these commonly used methods, a composite method that is based on the confidence intervals of the negative binomial distribution is proposed, which benefits from the relationship between negative binomial distribution and negative binomial group distribution. Simulation studies are carried out to examine the performances of our methods. A real data example is also presented to illustrate the application of our method.     

Score tests for heterogeneity and overdispersion in zero‐inflated Poisson and binomial regression models

Hall (2000) has described zero‐inflated Poisson and binomial regression models that include random effects to account for excess zeros and additional sources of heterogeneity in the data. The authors of the present paper propose a general score test for the null hypothesis that variance components associated with these random effects are zero. For a zero‐inflated Poisson model with random intercept, the new test reduces to an alternative to the overdispersion test of Ridout, Demério & Hinde (2001). The authors also examine their general test in the special case of the zero‐inflated binomial model with random intercept and propose an overdispersion test in that context which is based on a beta‐binomial alternative.     

On confidence limits for the difference of two binomial parameters

In the present article we suggest two new methods for calculating approximate confidence limits for the differences of the two binomial parameters. Different methods for determining the confidence interval are compared.     

Improved approximations for the inverse moments of the positive binomial distribution

Several alternatives to the most common approximation to the inverse moments of the positive binomial distribution are obtained. The method is based on equating moments and gives considerably better approximations for some values of the parameters.     

An EM algorithm for estimating negative binomial parameters

An EM algorithm is proposed for computing estimates of parameters of the negative bi-nomial distribution; the algorithm does not involve further iterations in the M-step, in contrast with the one given in Schader & Schmid (1985). The approach can be applied to the corresponding problem in the logarithmic series distribution. The convergence of the proposed scheme is investigated by simulation, the observed Fisher information is derivedand numerical examples based on real data are presented.     

Generalized binomial distributions

In many cases where the binomial dismbution fails to apply to real world data it is because of more variability in the data than can be explained by that dismbution. Several authors have proposed models that are useful in explaining extra-binomial variation. In this paper we point out a characterization of sequences of exchangeable Bernoulli random variables which can be used to develop models which show more variability than the binomial. We give sufficient conditions which will yield such models and show how existig models can be combined to generate further models. The usefulness of some of these models is illustrated by fitting them to sets of real data.     

Zero-inflated Poisson and negative binomial integer-valued GARCH models

Zero inflation means that the proportion of 0's of a model is greater than the proportion of 0's of the corresponding Poisson model, which is a common phenomenon in count data. To model the zero-inflated characteristic of time series of counts, we propose zero-inflated Poisson and negative binomial INGARCH models, which are useful and flexible generalizations of the Poisson and negative binomial INGARCH models, respectively. The stationarity conditions and the autocorrelation function are given. Based on the EM algorithm, the estimating procedure is simple and easy to be implemented. A simulation study shows that the estimation method is accurate and reliable as long as the sample size is reasonably large. A real data example leads to superior performance of the proposed models compared with other competitive models in the literature.     

A generalization of the binomial distribution

This paper presents a new departure in the generalization of the binomial distribution by adopting the assumption that the underlying Bernoulli trials take on the values α or β where α < β, rather than the conventional values 0 or 1. The adoption of this more general assumption renders the binomial distribution a four-parameter distribution of the form B(n,p,α,β), and requires the generalization of Romanovsky's (1923) reduction formula for central moments. This paper assesses the usefulness of B(n,p,α,β), and its reduction formula, in the numerical analysis of two problems of interest to decision theorists.     

“Faulty inspeciton” distributions

The distribution of the number of items observed to be defective in samples from a finite population. When detection of defectiveness is not certain, is obtained. The distribution of waiting time all a specified number of defective items is observed is also considered.

     

Testing the linearity of negative binomial regression models

The negative binomial (NB) is frequently used to model overdispersed Poisson count data. To study the effect of a continuous covariate of interest in an NB model, a flexible procedure is used to model the covariate effect by fixed-knot cubic basis-splines or B-splines with a second-order difference penalty on the adjacent B-spline coefficients to avoid undersmoothing. A penalized likelihood is used to estimate parameters of the model. A penalized likelihood ratio test statistic is constructed for the null hypothesis of the linearity of the continuous covariate effect. When the number of knots is fixed, its limiting null distribution is the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom. The smoothing parameter value is determined by setting a specified value equal to the asymptotic expectation of the test statistic under the null hypothesis. The power performance of the proposed test is studied with simulation experiments.     

Expectation identity for the binomial distribution and its application in the calculations of high-order binomial moments

ABSTRACT

For the exponential families normal, gamma, beta, Poisson, and negative binomial, there exists an expectation identity for each of the family. For the binomial family, we discover an expectation identity, which is useful in analytical calculations of its high-order moments.     

A note on the determination of sample sizes for hypergeometric distributions

Sample size determination for testing the hypothesis of equality of two proportions against an alternative with specified type I and type II error probabilities is considered for two finite populations. When two finite populations involved are quite different in sizes, the equal size assumption may not be appropriate. In this paper, we impose a balanced sampling condition to determine the necessary samples taken without replacement from the finite populations. It is found that our solution requires smaller samples as compared to those using binomial distributions. Furthermore, our solution is consistent with the sampling with replacement or when population size is large. Finally, three examples are given to show the application of the derived sample size formula.     

A note on LeCam’s bound for the distance between the Poisson binomial and the Poisson distribution

n possibly different success probabilities p ₁, p ₂, ..., p _n is frequently approximated by a Poisson distribution with parameter λ = p ₁ + p ₂ + ... + p _n . LeCam's bound p ² ₁ + p ² ₂ + ... + p _n ² for the total variation distance between both distributions is particularly useful provided the success probabilities are small. The paper presents an improved version of LeCam's bound if a generalized d-dimensional Poisson binomial distribution is to be approximated by a compound Poisson distribution. Received: May 10, 2000; revised version: January 15, 2001     

An approximation of the inverse moments of the positive hypergeometric distribution

The negative moments of the positive hyper geometric distribution are often approximated by the inverse of the positive moments of this distribution. In this paper, a suitable approximation to the positive hypergeometric distribution is used to obtain the negative moments.     

On the differences of two generalized negative binomial variates

The generalized negative binomial (GNB) distribution was defined by Jain and Consul (SIAM J. Appl. Math., 21 (1971)) and was obtained as a particular family of Lagrangian distributions by Consul and Shenton (SIAM J. Appl. Math., 23 (1973)). Consul and Shenton also gave the probability generating function (p.g.f.) and proved many properties of the GNBD. Consul and Gupta (SIAM J. Appl. Math., 39 (1980)) proved that the parameter β must be either zero or 1≤ β ≤ θ^-1 for the GNBD to be a true probability distribution and proved some other properties. Numerous applications and properties of this model have been studied by various researchers. Considering two independent GNB variates X and Y, with parameters (m,β,θ) and (n,β,θ) respectively, the probability distribuition of D = Y-X and its p.g.f. and cumulant generating function have been obtained. A recurrence relation between the cumulants has been established and the first four cumulants, β₁ and β₂ have been derived. Also some moments of the absolute difference |Y-X| have been obtained.     

On the beta-correlated binomial (bcb) distribution - a three parameter generalization of the binomial distribution

A modification of the beta-correlated binomial (BCB) distribution of Paul (1985) is proposed. This modification over- comes some theoretical difficulties encountered in the original BCB distribution.     

Transform methods for the hypergeometric distribution

Two new methods for computing with hypergeometric distributions on lattice points are presented. One uses Fourier analysis, and the other uses Gröbner bases in the Weyl algebra. Both are very general and apply to log-linear models that are graphical or non-graphical.     

Conservative confidence intervals for a single parameter

The method of constructing confidence intervals from hypothesis tests is studied in the case in which there is a single unknown parameter and is proved to provide confidence intervals with coverage probability that is at least the nominal level. The confidence intervals obtained by the method in several different contexts are seen to compare favorably with confidence intervals obtained by traditional methods. The traditional intervals are seen to have coverage probability less than the nominal level in several instances, This method can be applied to all confidence interval problems and reduces to the traditional method when an exact pivotal statistic is known.