Associated Natural Exponential Families and Elliptic Functions

This paper studies the variance functions of the natural exponential families (NEF) on the real line of the form $$(Am^4+Bm^2+C)^{1/2}$$ ( A m 4 + B m 2 + C ) 1 / 2 where m denoting the mean. Surprisingly enough, most of them are discrete families concentrated on $$\lambda \mathbb {Z}$$ λ Z for some constant $$\lambda $$ 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 $$V_F(m)=V_G(m\sqrt{-1})$$ 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 $$m(Cm^4+Bm^2+A)^{1/2}.$$ m ( C m 4 + B m 2 + A ) 1 / 2 . They are easier to study and they are concentrated on $$a\mathbb {N}$$ a N .