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Generalized variance and exponential families. (English) Zbl 0945.62017
Summary: 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 \(F(\mu)\).

MSC:
62E10 Characterization and structure theory of statistical distributions
60E05 Probability distributions: general theory
62H05 Characterization and structure theory for multivariate probability distributions; copulas
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