## Abstract

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 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 (X_{1},…,X_{n}); n ≥ 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.

Original language | English |
---|---|

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