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February 1999 Generalized variance and exponential families
Abdelhamid Hassairi
Ann. Statist. 27(1): 374-385 (February 1999). DOI: 10.1214/aos/1018031116

Abstract

Let $\mu$ be a positive measure on $\mathbb{R}^d$ and let $F(\mu) = \{P(\theta,\mu); \theta \in \Theta\}$ be the natural exponential family generated by $\mu$. The aim of this paper is to show that if $\mu$ is infinitely divisible then the generalized variance of $\mu$, .i.e., the determinant of the covariance operator of $P(\theta,\mu)$, is the Laplace transform of some positive measure $\rho(\mu)$ on $mathbb{R}^d$. We then investigate the effect of the transformation $\mu \to \rho(\mu)$ and its implications for the skewness vector and the conjugate prior distribution families of $F(\mu)$. .

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Abdelhamid Hassairi. "Generalized variance and exponential families." Ann. Statist. 27 (1) 374 - 385, February 1999. https://doi.org/10.1214/aos/1018031116

Information

Published: February 1999
First available in Project Euclid: 5 April 2002

zbMATH: 0945.62017
MathSciNet: MR1701116
Digital Object Identifier: 10.1214/aos/1018031116

Subjects:
Primary: 62E10
Secondary: 62H05

Rights: Copyright © 1999 Institute of Mathematical Statistics

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Vol.27 • No. 1 • February 1999
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