Reproductive Exponential Families

Abstract

Consider a full and steep exponential model M with model function a(θ)b(x)exp{θ · t(x)} and a sample x1, ⋯, xn from M. Let t̄ = {t(x1) + ⋯ + t(xn)}/n and let t̄ = (t̄1, t̄2) be a partition of the canonical statistic t̄. We say that M is reproductive in t2 if there exists a function H independent of n such that for every n the marginal model for t̄2 is exponential with nθ as canonical parameter and (H(t̄2), t̄2) as canonical statistic. Furthermore we call M strongly reproductive if these marginal models are all contained in that for n = 1. Conditions for these properties to hold are discussed. Reproductive exponential models are shown to allow of a decomposition theorem analogous to the standard decomposition theorem for χ2-distributed quadratic forms in normal variates. A number of new exponential models are adduced that illustrate the concepts and also seem of some independent interest. In particular, a combination of the inverse Gaussian distributions and the Gaussian distributions is discussed in detail.