A Classification of Bivariate Negative Binomial Distributions

Abstract

The class of bivariate negative binomial distributions is defined as that of discrete distributions with both marginal distributions being negative binomial. A subclass are bivariate negative multinomial distributions with probability generating function (pgf) (1 + p − apt ₁ − bpt ₂ ) ^-k, a + b = 1 (Johnson, Kotz & Balakrishnan, 1997). A trivial type is that of independence, i.e. that with pgf (1 +p₁ − p ₁t_l)^−k ₁ (1 +p ₂ −p ₂ t ₂ ) ^−k ₂ Derived types are bivariate negative binomial distributions of random variables X ₁, X ₁ ± X ₂. If a bivariate distribution is represented as an (infinite) matrix its probability mass function (pmf) can be described by the initial vector of the marginal row probabilities and the transition matrix of the conditional row probabilities. Other types are obtained from products of stochastic matrices. The joint distribution of random variables X ₁ ± X ₁₂, X ₂ ± X ₁₂ is an example. A bivariate negative binomial distribution that is not multinomial is obtained from a randomly-stopped sums distribution: the product of a Poisson-logarithmic series distribution with only one marginal distribution being negative binomial, postmultiplied by the stochastic matrix of its transpose. Many other types of negative binomial, not multinomial distributions can be constructed in this way. While the forementioned types are parametric non-parametric distributions could also be generated from special singular value decompositions. If marginal right-censored negative distributions are admitted two different adjustment methods of contingency tables due to Deming and Stephan (1940) and Stephan (1942) can be applied, one of them known as Iterative Proportional Fitting (IPF).

The scope of the present paper is a classification of parametric bivariate negative binomial distributions according to structural features not an attempt to create new types.