When does the minimum of a sample of an exponential family belong to an exponential family? (English) Zbl 1338.62042

Bar-Lev, Shaul K.; Letac, Gérard

Electron. Commun. Probab. 21, Paper No. 6, 8 p. (2016).

Summary: It is well known that if \(({X}_{1},\ldots,{X}_{n})\) are i.i.d. r.v.’s taken from either the exponential distribution or the geometric one, then the distribution of \(\min({X}_{1},\ldots,{X}_{n})\) 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 \(({X}_{1},\ldots,{X}_{n})\), \(n \geq 2\), taken from \(F\) and let \(Y =\min({X}_{1},\ldots,{X}_{n})\). 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.

MSC:

62E10

Characterization and structure theory of statistical distributions

Keywords:

exponential family; exponential distribution; geometric distribution; order statistics; Radon measure