The Annals of Statistics

Generalized variance and exponential families

Abdelhamid Hassairi

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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

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

First available in Project Euclid: 5 April 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Natural exponential family variance function generalized variance skewness vector.


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

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  • CASALIS, M. 1996. The 2 d 4 simple quadratic natural exponential families on. Ann. Statist. 24 1828 1854. Z.
  • CONSONNI, G. and VERONESE, P. 1992. Conjugate priors for exponential families having quadratic variance function. J. Amer. Statist. Assoc. 87 1123 1127. Z.
  • DIACONIS, P. and YLVISAKER, D. 1979. Conjugate priors for exponential families. Ann. Statist. 7 269 281. Z.
  • 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, ´ Imperial 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. ´ ´ Z. Z.
  • 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. ´ Z.
  • KOKONENDJI, C. and SESHADRI, V. 1994. The Lindsay transform of natural exponential families. Canad. J. Statist. 22 259 272. Z.
  • KOKONENDJI, C. and SESHADRI, V. 1996. On the determinant of the second derivative of the Laplace transform. Ann. Statist. 24 1813 1827. Z.
  • LETAC, G. 1992. Lectures on Natural Exponential Families and Their Variance-Functions.
  • IMPA, Rio de Janeiro.
  • LETAC, G. and MORA, M. 1990. Natural real exponential families with cubic variance functions. Ann. Statist. 18 1 37. Z.
  • LINDSAY, B. G. 1989. On the determinant of moment matrices. Ann. Statist. 17 711 721. Z.
  • MORRIS, C. N. 1982. Natural exponential families with quadratic variance functions. Ann. Statist. 10 65 80. Z.
  • POLYA, G. and SZEGO, G. 1972. Problems and Theorems in Analysis 1. Springer, Berlin. ´ ¨ Z.
  • WILKS, S. S. 1932. Certain generalizations in the analysis of variance. Biometrika 24 471 794.