A note on inconsistent families of discrete multivariate distributions

Sugata Ghosh,Subhajit Dutta,Marc G Genton

doi:10.1186/s40488-017-0061-8

Abstract

We construct a d-dimensional discrete multivariate distribution for which any proper subset of its components belongs to a specific family of distributions. However, the joint d-dimensional distribution fails to belong to that family and in other words, it is ‘inconsistent’ with the distribution of these subsets. We also address preservation of this ‘inconsistency’ property for the symmetric Binomial distribution, and some discrete distributions arising from the multivariate discrete normal distribution.

Highlights

1 Introduction If a multivariate distribution is parametrically specified, often the lower-dimensional marginals follow the same distribution of an appropriate dimension
Characterization problems for discrete distributions using conditional distributions and their expectations have been investigated by several authors; see, e.g., Dahiya and Korwar (1977), Ruiz and Navarro (1995), and Nguyen et al (1996)
The paper by Conway (1979) discussed some additional properties and derived appropriate relationships for such systems, while a method of constructing multivariate distributions with specified univariate marginals and a given correlation matrix was studied by Cuadras (1992)

Summary

Introduction

If a multivariate distribution is parametrically specified, often the lower-dimensional marginals follow the same distribution of an appropriate dimension. We construct a discrete multivariate distribution (see, e.g., Johnson et al (1997)), all of whose lower-dimensional marginals follow the same symmetric distribution but the joint distribution does not conform to that pattern and has a different distribution. Consider a random variable U that follows a Binomial distribution with parameters n =

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Conclusion