Generalized variance and exponential families

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 $F(\mu)$. .