zbMATH — the first resource for mathematics

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 \(F(\mu)\).

62E10 Characterization and structure theory of statistical distributions
60E05 Probability distributions: general theory
62H05 Characterization and structure theory for multivariate probability distributions; copulas
Full Text: DOI
[1] CASALIS, M. 1996. The 2 d 4 simple quadratic natural exponential families on. Ann. Statist. 24 1828 1854. Z. · Zbl 0867.62042
[2] CONSONNI, G. and VERONESE, P. 1992. Conjugate priors for exponential families having quadratic variance function. J. Amer. Statist. Assoc. 87 1123 1127. Z. JSTOR: · Zbl 0764.62027
[3] DIACONIS, P. and YLVISAKER, D. 1979. Conjugate priors for exponential families. Ann. Statist. 7 269 281. Z. · Zbl 0405.62011
[4] GIKHMAN, I. I. and SKOROHOD, A. V. 1973. The Theory of Stochastic Processes 2. Springer, New York. Z. GUTIERREZ-PENA, E. 1995. Bayesian topics relating to the exponential family. Ph.D. thesis, Ímperial College, London. Z. n HASSAIRI, A. 1992. La classification des familles exponentielles naturelles sur par l’action du groupe lineaire de n 1. C.R. Acad. Sci. Paris Ser. I 315 207 210. \' Ź. Z. · Zbl 0752.62011
[5] HASSAIRI, A. 1993. Les d 3 G-orbites de la classe de Morris-Mora des familles exponentielles de d. C.R. Acad. Sci. Paris Ser. I 317 887 890. Ź. · Zbl 0799.62016
[6] KOKONENDJI, C. and SESHADRI, V. 1994. The Lindsay transform of natural exponential families. Canad. J. Statist. 22 259 272. Z. JSTOR: · Zbl 0811.62022
[7] KOKONENDJI, C. and SESHADRI, V. 1996. On the determinant of the second derivative of the Laplace transform. Ann. Statist. 24 1813 1827. Z. · Zbl 0868.62047
[8] LETAC, G. 1992. Lectures on Natural Exponential Families and Their Variance-Functions. · Zbl 0983.62501
[9] IMPA, Rio de Janeiro.
[10] LETAC, G. and MORA, M. 1990. Natural real exponential families with cubic variance functions. Ann. Statist. 18 1 37. Z. · Zbl 0714.62010
[11] LINDSAY, B. G. 1989. On the determinant of moment matrices. Ann. Statist. 17 711 721. Z. · Zbl 0672.62062
[12] MORRIS, C. N. 1982. Natural exponential families with quadratic variance functions. Ann. Statist. 10 65 80. Z. · Zbl 0498.62015
[13] POLYA, G. and SZEGO, G. 1972. Problems and Theorems in Analysis 1. Springer, Berlin. \' \" Z.
[14] WILKS, S. S. 1932. Certain generalizations in the analysis of variance. Biometrika 24 471 794. · Zbl 0006.02301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.