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

Shaul K. Bar-Lev, Gérard Letac

Research output: Contribution to journalArticlepeer-review


It is well known that if (X1,…,Xn) are i.i.d. r.v.’s taken from either the exponential distribution or the geometric one, then the distribution of min(X1,…,Xn) 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,…,Xn); n ≥ 2; taken from F and let Y = min(X1,…,Xn). 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.

Original languageEnglish
Article number6
JournalElectronic Communications in Probability
StatePublished - 2016

Bibliographical note

Publisher Copyright:
© 2016, University of Washington. All rights reserved.


  • Exponential distribution
  • Exponential family
  • Geometric distribution
  • Order statistics
  • Radon measure

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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