Electronic Communications in Probability

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

Shaul K. Bar-Lev and Gérard Letac

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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.

Article information

Electron. Commun. Probab., Volume 21 (2016), paper no. 6, 8 pp.

Received: 3 August 2015
Accepted: 24 December 2015
First available in Project Euclid: 3 February 2016

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Zentralblatt MATH identifier

Primary: 62E10: Characterization and structure theory

exponential family exponential distribution geometric distribution order statistics Radon measure

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Bar-Lev, Shaul K.; Letac, Gérard. When does the minimum of a sample of an exponential family belong to an exponential family?. Electron. Commun. Probab. 21 (2016), paper no. 6, 8 pp. doi:10.1214/16-ECP4458. https://projecteuclid.org/euclid.ecp/1454514626

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