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September, 1983 Reproductive Exponential Families
O. Barndorff-Nielsen, P. Blaesild
Ann. Statist. 11(3): 770-782 (September, 1983). DOI: 10.1214/aos/1176346244

Abstract

Consider a full and steep exponential model $\mathscr{M}$ with model function $a(\theta)b(x)\exp\{\theta \cdot t(x)\}$ and a sample $x_1, \cdots, x_n$ from $\mathscr{M}$. Let $\bar{t} = \{t(x_1) + \cdots + t(x_n)\}/n$ and let $\bar{t} = (\bar{t}_1, \bar{t}_2)$ be a partition of the canonical statistic $\bar{t}$. We say that $\mathscr{M}$ is reproductive in $t_2$ if there exists a function $H$ independent of $n$ such that for every $n$ the marginal model for $\bar{t}_2$ is exponential with $n\theta$ as canonical parameter and $(H(\bar{t}_2), \bar{t}_2)$ as canonical statistic. Furthermore we call $\mathscr{M}$ strongly reproductive if these marginal models are all contained in that for $n = 1$. Conditions for these properties to hold are discussed. Reproductive exponential models are shown to allow of a decomposition theorem analogous to the standard decomposition theorem for $\chi^2$-distributed quadratic forms in normal variates. A number of new exponential models are adduced that illustrate the concepts and also seem of some independent interest. In particular, a combination of the inverse Gaussian distributions and the Gaussian distributions is discussed in detail.

Citation

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O. Barndorff-Nielsen. P. Blaesild. "Reproductive Exponential Families." Ann. Statist. 11 (3) 770 - 782, September, 1983. https://doi.org/10.1214/aos/1176346244

Information

Published: September, 1983
First available in Project Euclid: 12 April 2007

zbMATH: 0529.62016
MathSciNet: MR707928
Digital Object Identifier: 10.1214/aos/1176346244

Subjects:
Primary: 62E15
Secondary: 62F99

Rights: Copyright © 1983 Institute of Mathematical Statistics

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Vol.11 • No. 3 • September, 1983
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