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 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 aN.

Two further probabilistic applications of Bessel functions

“…Whereas a general solution to this problem does not seem to exist, several examples have been given in [14]. See also Section 3 in [12], which uses the different terminology associated pair. The prototype is the standard normal distribution with the van Dantzig pair (e −t 2 /2 , e −t 2 /2 ), which is called self-reciprocal because its two components are equal.…”

The Full Tails Gamma Distribution Applied to Model Extreme Values

“…Given σ fixed, the family (2.1) for α ≤ 0 appears as the full exponential model generated from a canonical statistic (x, log (σ + x)), see Barndorff-Nielsen (1978), Brown (1986), Letac (1992).…”

Resolution of a conjecture on variance functions for one-parameter natural exponential families

“…Therefore, identifying which f ∈ G D are VFs of NFEs is of importance. For few of these results, see, e.g., Bar-Lev et al (1991); ; Letac (2016).…”