Abstract
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 language | English |
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Article number | 6 |
Journal | Electronic Communications in Probability |
Volume | 21 |
DOIs | |
State | Published - 2016 |
Bibliographical note
Publisher Copyright:© 2016, University of Washington. All rights reserved.
Keywords
- Exponential distribution
- Exponential family
- Geometric distribution
- Order statistics
- Radon measure
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty