When does the minimum of a sample of an exponential family belong to an exponential family?

Abstract

It is well known that if $({X}_{1},...,{X}_{n})$ are i.i.d. r.v.'s taken from either the exponential distribution or the geometric one, then the distribution of $\min({X}_{1},...,{X}_{n})$ is, with a change of parameter, is also exponential or geometric, respectively. In this note we prove the following result. Let $F$ be a natural exponential family (NEF) on $\mathbb{R}$ generated by an arbitrary positive Radon measure $\mu$ (not necessarily confined to the Lebesgue or counting measures on $\mathbb{R}$). Consider $n$ i.i.d. r.v.'s $({X}_{1},...,{X}_{n})$, $n \in 2$, taken from $F$ and let $Y =\min({X}_{1},...,{X}_{n})$. We prove that the family $G$ of distributions induced by $Y$ constitutes an NEF if and only if, up to an affine transformation, $F$ is the family of either the exponential distributions or the geometric distributions. The proof of such a result is rather intricate and probabilistic in nature.