Associated Natural Exponential Families and Elliptic Functions

Abstract

This paper studies the variance functions of the natural exponential families (NEF) on the real line of the form \((Am^4+Bm^2+C)^{1/2}\) where m denoting the mean. Surprisingly enough, most of them are discrete families concentrated on \(\lambda \mathbb {Z}\) for some constant \(\lambda \) and the Laplace transform of their elements are expressed by elliptic functions. The concept of association of two NEF is an auxiliary tool for their study: two families F and G are associated if they are generated by symmetric probabilities and if the analytic continuations of their variance functions satisfy \(V_F(m)=V_G(m\sqrt{-1})\). We give some properties of the association before its application to these elliptic NEF. The paper is completed by the study of NEF with variance functions \(m(Cm^4+Bm^2+A)^{1/2}.\) They are easier to study and they are concentrated on \(a\mathbb {N}\).