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

It is well known that if (X₁,…,X_n) are i.i.d. r.v.’s taken from either the exponential distribution or the geometric one, then the distribution of min(X₁,…,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 ℝ generated by an arbitrary positive Radon measure m (not necessarily confined to the Lebesgue or counting measures on ℝ). Consider n i.i.d. r.v.’s (X₁,…,X_n); n ≥ 2; taken from F and let Y = min(X₁,…,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.