On bivariate generalized binomial and negative binomial distributions

The present paper deals with two types of generalized general binomial (binomial or negative binomial) distributions: (i) a univariate general binomial generalized by a bivariate distribution and (ii) a bivariate general binomial generalized by two independent univariate distributions. The probabilities, moments, conditional distributions and regression functions for these distributions are obtained in terms of bipartitional polynomials. Moreover recurrence relations for the probabilities and moments, independent of the bipartitional polynomials, are given. Finally these general results are applied to the (i) Binomial-Bivariate Poisson and (ii) Bivariate Binomial-Poissons distributions.     

Some notes on the non-central negative binomial distribution

Summary In this paper we shall record some facts and further examine certain properties of the non-central negative binomial (NNB) distribution (Laguerre series distribution of Gurland, Chen and Hernandez 1983). We consider, among others, a stochastic formulation (birth-and-death process), the series expansion of the probability distribution and the corresponding series expansion of a generalized exponential distribution (Ong/Lee 1986), and the connection of the NNB distribution with the non-central beta, gamma and non-central gamma distributions. A four-parameter version of the NNB distribution is also presented.     

Bivariate non-central negative binomial distribution: Another generalisation

Summary Another bivariate generalisation (Type V) of the non-central negative binomial distribution is considered. This generalisation is constructed (i) as a latent structure model; (ii) as an extension of an accident proneness model investigated by Edwards/Gurland (1961); and (iii) as a reversible stochastic counter model. The third construction gives, as a result, an apparently new formulation of the Edwards/Gurland model. The probabilities, moments, recurrence formulas and some properties are given. An application to the data used by Holgate (1966) is considered.     

A negative binomial thinning-based bivariate INAR(1) process

This article considers a bivariate INAR(1) process based on an extension of the negative binomial thinning operator by prespecifying the distribution of the innovations. The dependence is introduced through the innovation components. The existence, uniqueness, strict stationarity, ergodicity, and some probabilistic properties of the process are derived. The estimation methods of conditional least squares and conditional maximum likelihood are considered. Some numerical results of the estimates are presented by simulation study. An application to crime data set is provided.     

On some properties of the linear function Poisson binomial and negative binomial distributions

Summary A general model in fluctuations of sums of random variables leading, under certain assumptions, to each of the generalized and linear function Poisson, binomial and negative binomial distributions is presented. Moreover the generating functions and the factorial moments of the linear function Poisson, binomial and negative binomial distributions are obtained in close forms and certain distributional properties are discussed.     

Compound gamma bivariate distributions

Summary Bivariate distributions, which may be of special relevance to the lifetimes of two components of a system, are derived using the following approach. As the two components are part of one system and therefore exposed to similar conditions of service, there will be similarity between their lifetimes that is not shared by components belonging to different systems. The lifetime distribution for a given system is assumed to be Gamma in form (this includes the exponential as a special case; extension to the Stacey distribution, which includes the Weibull distribution, is straightforward). The scale parameter of this distribution is itself a random variable, with a Gamma distribution. We thus obtain what might be termed a compound Gamma-Gamma bivariate distribution. The cumulative distribution function of this may be expressed in terms of one of the double hypergeometric functions of Appell.Generalised hypergeometric functions play an important part in this paper, and one of Saran's triple hypergeometric functions is obtained when generalising the above model to permit the scale parameters of the distributions for the two components to be correlated, rather than identical.Work started while the author was with the Transport Studies Group, University College London.     

Bounds for the difference between median and mean of beta and negative binomial distributions

Summary We obtain bounds for the difference between the median and mean of the beta and negative binomial distributions using elementary methods.     

A unified approach to bivariate discrete distributions

Here we introduce a bivariate generalized hypergeometric factorial moment distribution (BGHFMD) through its probability generating function (p.g.f.) whose marginal distributions are the generalized hypergeometric factorial moment distributions introduced by Kemp and Kemp (Bull Int Stat Inst 43:336–338,1969). Well-known bivariate versions of distributions such as binomial, negative binomial and Poisson are special cases of this distribution. A genesis of the distribution and explicit closed form expressions for the probability mass function of the BGHFMD, its factorial moments and the p.g.f.’s of its conditional distributions are derived here. Certain recurrence relations for probabilities, moments and factorial moments of the bivariate distribution are also established.     

Conditional distributions and the bivariate normal distribution

Summary An alaysis of the extent to which conditional distributions of a bivariate vector characterize bivariate normality is given.     

A characterization of the binomial and negative binomial distribution by regression of products

Summary Products of independent random variables with different shifts are considered. The binomial and negative binomial populations are characterized by proportionality of regressions of these products with respect to residuals.     

Bounds for median and 50 percetage point of binomial and negative binomial distribution

Inequalities for comparing the binomial distribution function/operating characteristic function with value 0.5 have been established long ago. It is a trivial task to infer from these inequalities bounds for the 50 percentage point of the binomial operating characteristic function. However, it is also possible to use these inequalities to establish good bounds for the median of the binomial distribution function. By a relation of binomial distribution function/operating characteristic function and negative binomial distribution function/operating characteristic function we can get bounds for the respective parameters of negative binomial distribution, too.     

A new family of bivariate max-infinitely divisible distributions

In this article we discuss the asymptotic behaviour of the componentwise maxima for a specific bivariate triangular array. Its components are given in terms of linear transformations of bivariate generalised symmetrised Dirichlet random vectors introduced in Fang and Fang (Statistical inference in elliptically contoured and related distributions. Allerton Press, New York, 1990). We show that the componentwise maxima of such triangular arrays is attracted by a bivariate max-infinitely divisible distribution function, provided that the associated random radius is in the Weibull max-domain of attraction.     

Some general results for moments in bivariate distributions

The purpose of this paper is to present a closed formula to compute the moments of a general function from the knowledge of its bivariate survival function. The result is derived by utilizing an integration by parts formula for two variables, which is not readily available in the literature. Many of the existing results are obtained as special cases. Finally, two examples are presented to illustrate the results. In both the examples, mixed moments as well as moments for the series system and parallel system are obtained. The integration by parts formula in two variables, derived here, is of interest in its own right and we hope that it will be useful in other investigations. The integration by parts formula in two variables is derived as a special case of a general formula in n variables.     

On three and five parameter bivariate beta distributions

Summary In this paper two bivariate beta distributions have been studied. The five parameter bivariate beta distribution is derived from the Morgenstern-system of curves while the three parameter distribution is the bivariate Dirichlet distribution. In both cases the distributions of the product and the quotient of random variables are derived and other properties are also studied.     

On truncated bivariate discrete distributions: A unified treatment

A unified treatment of three types of zero class truncation for bivariate discrete distributions is presented. Using the probability generating function approach, various properties of the truncated distributions are examined in association with the corresponding properties of the initial complete form of the distribution. Expressions for moments and conditional distributions are also obtained. Bivariate versions of the Thomas and the Intervened Poisson distributions are introduced and used as illustrative examples. Received November 2000/Revised March 2002     

Characterizations of bivariate distributions using concomitants of record values

In this paper, we consider a family of bivariate distributions which is a generalization of the Morgenstern family of bivariate distributions. We have derived some properties of concomitants of record values which characterize this generalized class of distributions. The role of concomitants of record values in the unique determination of the parent bivariate distribution has been established. We have also derived properties of concomitants of record values which characterize each of the following families viz Morgenstern family, bivariate Pareto family and a generalized Gumbel’s family of bivariate distributions. Some applications of the characterization results are discussed and important conclusions based on the characterization results are drawn.     

Common factors in conditional distributions for bivariate time series

A definition for a common factor for bivariate time series is suggested by considering the decomposition of the conditional density into the product of the marginals and the copula, with the conditioning variable being a common factor if it does not directly enter the copula. We show the links between this definition and the idea of a common factor as a dominant feature in standard linear representations. An application using a business cycle indicator as the common factor in the relationship between U.S. income and consumption found that both series held the factor in their marginals but not in the copula.     

Bounds for the median of the negative binomial distribution

Summary Bounds are obtained for the median of the negative binomial distribution which are valid for all possible parameter values of the distribution when the median is defined as inf {x: P(X≤x)≥1/2}.     

Highest posterior mass prediction intervals for binomial and poisson distributions

The problems of constructing prediction intervals (PIs) for the binomial and Poisson distributions are considered. New highest posterior mass (HPM) PIs based on fiducial approach are proposed. Other fiducial PIs, an exact PI and approximate PIs are reviewed and compared with the HPM-PIs. Exact coverage studies and expected widths of prediction intervals show that the new prediction intervals are less conservative than other fiducial PIs and comparable with the approximate one based on the joint sampling approach for the binomial case. For the Poisson case, the HPM-PIs are better than the other PIs in terms of coverage probabilities and precision. The methods are illustrated using some practical examples.     

Characterizations of negative multinomial distributions based on conditional distributions

In this paper characterizations of negative multinomial distributions based on conditional distributions have been studied.