Associated natural exponential families and elliptic functions

Gérard Letac (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

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 λ Z 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 auxilliary 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 √ −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 N.

Mots clés: Variance functions; exponential dispersion models; function ℘ of Weierstrass

Domaines: Mathématiques [math]

Gérard Letac. Associated natural exponential families and elliptic functions. Conference on Probability, statistics and their applications, Jun 2015, Aarhus, Denmark. ⟨hal-01977671⟩