## The Annals of Statistics

### Generalized variance and exponential families

Abdelhamid Hassairi

#### 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 $F(\mu)$. .

#### Article information

Source
Ann. Statist., Volume 27, Number 1 (1999), 374-385.

Dates
First available in Project Euclid: 5 April 2002

https://projecteuclid.org/euclid.aos/1018031116

Digital Object Identifier
doi:10.1214/aos/1018031116

Mathematical Reviews number (MathSciNet)
MR1701116

Zentralblatt MATH identifier
0945.62017

Subjects
Primary: 62E10: Characterization and structure theory
Secondary: 62H05: Characterization and structure theory

#### Citation

Hassairi, Abdelhamid. Generalized variance and exponential families. Ann. Statist. 27 (1999), no. 1, 374--385. doi:10.1214/aos/1018031116. https://projecteuclid.org/euclid.aos/1018031116

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