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 (Am4 + Bm2 +C)1/2 where m denoting the mean. Surprisingly enough, most of them are discrete families concentrated on λℤ for some constant λ 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 VF (m) = VG(m√-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(Cm4+ Bm2+ A)1/2. They are easier to study and they are concentrated on aℕ.