L'action quadratique du groupe des homographies sur les familles exponentielles réelles

Abstract

We define a quadratic action of SL(2,ℝ) by Mœbius transforms h(x) = (ax + b)/ (cx + d) on the natural exponential families (NEF) on ℝ, which changes the mean function k′ of the NEF F to the new mean function h(k′), associated with the new NEF denoted by h(F). The variance function of h(F) is (cm + d) 2V F(h(m)). When z → k′ (z) or z → ak′ (a log z) happens to be a Pick function, h(F) can be easily described. We prove that certain cubic NEF belong to this type, which then leads us to a classification of the variance functions P(m)/m, where P has degree ≤ 3.