Joint Probability Distributions in the Meixner Classes

Abstract

Summary The classes of statistical distributions, for which exp [xu(t)]/E(exp [xu(t)]) generates a system of orthogonal polynomials, have been characterized by Meixner (1934). His proof is now simplified. The theory is applied to obtain joint distributions in the normal, binomial, gamma and Poisson Meixner classes. The pre-eminent importance of those joint distributions with the polynomial biorthogonal property is established. Their simple regression properties are demonstrated. Some general theorems on convolutions of the biorthogonal distributions are proved.