Multivariate Negative Binomial Distribution Research Articles

A note on the multivariate negative binomial distributions of order K

A useful recurrence relation for calculating the probabilities of the multivariare negative binomial distrtibution of order k is obtained. Analogous recurrences are also established for the type I, type II and extended multivariate negative binomial distributions of the same order. The present paper extends to the multivariate case the work of Philippou and Antzoulakos (1989) on calculating the probabilities of the multi-parameter negative binomial distribution of order k.

Multivariate Polya and Inverse Polya Distributions of Orderk

Multivariate Polya and inverse Polya distributions of order k are derived by means of generalized urn models and by compounding the type II multinomial and multivariate negative binomial distributions of order k of PHILIPPOU, ANTZOULAKOS and TRIPSIANNIS (1990, 1988), respectively, with the Dirichlet distribution. It is noted that the above two distributions include as special cases a multivariate hypergeometric distribution of order k, a negative one, an inverse one, a negative inverse one and a discrete uniform of the same order. The probability generating functions, means, variances and covariances of the new distributions are obtained and five asymptotic results are established relating them to the above-mentioned multinomial and multivariate negative binomial distributions of order k, and to the type II negative binomial and the type I multivariate Poisson distributions of order k of PHILIPPOU (1983), and PHILIPPOU, ANTZOULAKOS and TRIPSIAN-NIS (1988), respectively. Potential applications are also indicated. The present paper extends to the multivariate case the work of PHILIPPOU, TRIPSIANNIS and ANTZOULAKOS (1989) on Polya and inverse Polya distributions of order k..

CLASSES OF MULTIVARIATE EXPONENTIAL AND MULTIAVARIATE GEOMETRIC DISTRIBUTIONS DERIVED FROM MARKOV PROCESSES

ABSTRACTWe define a class of multivariate exponential distributions as the distributions of occupancy times in upwards skip‐free Markov processes in continuous time. These distributions are infinitely divisible, and the multivariate gamma class defined by convolutions and fractions is a substantial generalization of the class defined by Johnson & Kotz (1972). Parallel classes of multivariate geometric and multivariate negative binomial distributions are constructed from occupancy times in ‘instant’ upwards skip‐free Markov chains. Maximum likelihood estimation and times series applications are discussed.

Multivariate distributions of order κ

Three multivariate distributions of order κ are introduced and studied. A multivariate negative binomial distribution of order κ is derived first, by means of an urn scheme, and two limiting cases of it are obtained next. They are, respectively, a multivariate Poisson distribution of order κ and a multivariate logarithmic series distribution of the same order. The probability generating functions, means variances and covariances of these distributions are obtained, and some further genesis schemes of them and interrelationships among them are also established. The present paper extends to the multivariate case the work of Philippou (1987) on multiparameter distributions of order κ. At the same time, several results of Aki (1985) on extended distributions of order κ are also generalized to the multivariate case.

A class of infinitely divisible multivariate negative binomial distributions

A particular class of multivariate negative binomial distributions has probability generating functions of the form | I− Q| α | I− QS| − α , where α>0 and S=diag( s 1, …, s n ). The main results of this paper concern characterizations of the infinitely divisible distributions of this class.

Discrete generalized Liouville-Type distribution and related multivariate distributions

Discrete analogue of the Liouville distribution is defined and is termed as Discrete Generalized Liouville-Type Distribution (DGL-TD). Firstly, properties in its factorial and ordinary moments are given. Then by finding the covariance matrix, partial and multiple correlations for DGL-TD are evaluated. Multinomial, multivariate negative binomial and multivariate log series distributions are shown as particular cases of this general distribution. The asymptotic distribution of the estimates of the parameters is also attempted.

Multivariate discrete exponential family of distributions and their properties

The paper generalizes the univariate discrete exponential family of distributions to the multivariate situation, and this generalization includes the multivariate power series distributions, the multivariate Lagrangian distributions, and the modified multivariate power-series distributions. This provides a unified approach for the study of these three classes of distributions. We obtain recurrence relations for moments and cumulants, and the maximum likelihood estimation for the discrete exponential family. These results are applied to some multivariate discrete distributions like the Lagrangian Poisson, Lagrangian (negative) multinomial, logarithmic series distributions and multivariate Lagrangian negative binomial distribution.

Definition and characterization of multivariate negative binomial distribution

The probability generating function ( pgf) of an n-variate negative binomial distribution is defined to be [ β( s 1,…, s n )] − k where β is a polynomial of degree n being linear in each s i and k > 0. This definition gives rise to two characterizations of negative binomial distributions. An n-variate linear exponential distribution with the probability function h( x 1,…, x n ) exp ( Σ i=1 n θ ix i) f(θ 1,…,θ n) is negative binomial if and only if its univariate marginals are negative binomial. Let S t , t = 1,…, m, be subsets of { s 1,…, s n } with empty ∩ t=1 m S t . Then an n-variate pgf is of a negative binomial if and only if for all s in S t being fixed the function is of the form of the pgf of a negative binomial in other s's and this is true for all t.

On a characterization of the dispersion matrix based on the properties of regression

This paper provides a partial solution to a problem posed by J. Neyman (1965) regarding the characterization of multivariate negative binomial distribution based on the properties of regression. It is shown that some of the properties of regression characterize the form of the nonsingular dispersion matrix of the parent distribution, which, interestingly enough, corresponds to only two types viz. those of positive and negative multivariate binomial distributions.

Orthogonal polynomials on the negative multinomial distribution

Orthogonal polynomials on the multivariate negative binomial distribution, (1 + Θ) −α−x( π j=0 p Θ j xj x j! ) Γ(α + x) Γ(α) where α > 0, Θ 1 > 0, x = Σ Θ i , x 0, x 1, …, x p = 0,1, … are constructed and their properties studied.

A generalised incomplete beta function and its application to multi-line stock control

The distribution function for the negative binomial distribution is known to be an incomplete beta function. Here, some of the properties of the family of distribution functions for multivariate negative binomial distributions are explored. These properties are then used in deriving the expected cost per unit time for a multi-line joint-reordering system with Poisson demands. Policies are considered for which the quantity of any particular line in stock is the same at the beginning of every cycle. A method which gives good approximations to the optimal values of these quantities is described.

A generalised incomplete beta function and its application to multi-line stock control

The distribution function for the negative binomial distribution is known to be an incomplete beta function. Here, some of the properties of the family of distribution functions for multivariate negative binomial distributions are explored. These properties are then used in deriving the expected cost per unit time for a multi-line joint-reordering system with Poisson demands. Policies are considered for which the quantity of any particular line in stock is the same at the beginning of every cycle. A method which gives good approximations to the optimal values of these quantities is described.

On the compound negative multinomial distribution and correlations among inversely sampled pollen counts

INTRODUCTION In the generalization of Bernoulli trials where we have k possible outcomes of each trial, k let the probability of the ith outcome in each trial be pi (i = 1,...,k) where Pi= 1. i=l For a fixed number of trials (n), the probability of exactly x1 occurrences of outcome 1, x2 of 2, ..., Xk of k is given by the multinomial distribution. If, however, the number of trials is not fixed in advance, but we continue to consider new trials until exactly r occurrences of the kth outcome (say) have been noted, then the probability of exactly xl occurrences of outcome 1, x2 of 2, ..., Xk1 of k 1, is given by the negative multinomial distribution. This latter procedure we shall refer to as 'inverse sampling' (cf. Tweedie, 1952). The negative multinomial distribution is clearly a generalization of the negative binomial. Just as this latter can be deduced from widely different models, so also can the negative multinomial. Thus Bates & Neyman (1952) arrive at this distribution (their multivariate negative binomial distribution) by considering the joint distribution of kIc-1 independent Poisson variables Xi (i = il ..., Ik-i) each with mean ai A. The parameter A is in turn regarded as a random variable A having a y(or type III) distribution. Integrating the joint frequency function of the X's and A with respect to the latter reveals that the marginal frequency function of the X's is that of the negative multinomial. (Using this approach r (r > 0) is the parameter of a y-distribution and is not restricted to integer values as in the model considered here.) Wishart (1949, p. 48) uses the term 'Pascal multinomial' for the case where r is an integer. The terms of the negative multinomial distribution are readily obtained as the coefficients k-1 of II t in the expansion of k-1 ]