## Abstract

Let F be a natural exponential family (NEF) generated by a measure μ and X = (X_{1}, ..., X_{n}) a random sample with a common distribution belonging to F. Consider the set of order statistics X_{(1)} ≤ X_{(2)} ≤ ⋯ ≤ X_{(n)} and let G_{r, n} denote the family of distributions induced by the r-th order statistic X_{(r)}, r = 1, ..., n. The main problem of the paper, namely, the closedness of NEF's under the formation of order statistics, can be posed as follows: for which NEF's F, the set of distributions G_{r, n} constitutes, for all n ∈ N and for some r ∈ {1, ..., n}, an NEF on R? If G_{r, n} is an NEF, we shall say that F is closed under the r-th order statistic. A comprehensive answer to this problem seems to be rather difficult when μ is an arbitrary measure. However, if μ is a continuous measure we show that if 1 < r < n, then G_{r, n} is not an NEF. The remaining cases r = 1 or r = n are equivalent under an appropriate affine transformation. For r = n we prove that G_{n, n} is an NEF if and only if F is the family of exponential distributions supported on R^{-}; or, equivalently, for r = 1, G_{1, n} is an NEF if and only if F is the family of exponential distributions supported on R^{+}.

Original language | English |
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Pages (from-to) | 2787-2792 |

Number of pages | 6 |

Journal | Statistics and Probability Letters |

Volume | 78 |

Issue number | 16 |

DOIs | |

State | Published - Nov 2008 |

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty