## Abstract

A polynomial Q = Q(X _{1},...,X _{n}) of degree m in independent identically distributed random variables with distribution function F is an unbiased estimator of a functional q(α _{1}(F), ... , α _{m}(F)), where q(u _{1},..., u _{m}) is a polynomial in u _{1},..., u _{m} and α _{j}(F) is the jth moment of F (assuming the necessary moment of F exists). It is shown that the relation E(Q {pipe} X _{1} + ... + Xn) = 0 holds if and only if q(α _{1}(θ), ... , α _{m}(θ)) ≡ 0, where α _{j}(θ) is the jth moment of the natural exponential family generated by F. This result, based on the fact that X _{1} + ... + X _{n} is a complete sufficient statistic for a parameter θ in a sample from a natural exponential family of distributions F _{θ}(x) = ∫ _{-∞} ^{x}e ^{θu-k(θ)}dF(u), explains why the distributions appearing as solutions of regression problems are the same as solutions of problems for natural exponential families though, at the first glance, the latter seem unrelated to the former.

Original language | English |
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Pages (from-to) | 201-206 |

Number of pages | 6 |

Journal | Mathematical Methods of Statistics |

Volume | 18 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2009 |

## Keywords

- Laha-Lukacs-Morris characterization
- Letac-Mora characterization
- exponential families
- variance function

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty