The \(2d+4\) simple quadratic natural exponential families on \(\mathbb{R}^ d\). (English) Zbl 0867.62042

Casalis, M.

Ann. Stat. 24, No. 4, 1828-1854 (1996).

Summary: The present paper describes all the natural exponential families on \(\mathbb{R}^d\) whose variance function is of the form \(V(m)= am\otimes m+B(m)+C\), with \(m\otimes m(\theta)= \langle \theta,m\rangle m\) and \(B\) linear in \(m\). There are \(2d+4\) types of such families, which are built from particular mixtures of families of normal, Poisson, gamma, hyperbolic on \(\mathbb{R}\) and (negative-) multinomial distributions. The proof of this result relies mainly on techniques used in the elementary theory of Lie algebras.

MSC:

62H05	Characterization and structure theory for multivariate probability distributions; copulas
62E10	Characterization and structure theory of statistical distributions
60E10	Characteristic functions; other transforms

Keywords:

simple quadratic natural exponential families; variance functions; Morris class; natural exponential families; mixtures; gamma; hyperbolic; multinomial distributions; Lie algebras; normal; Poisson

References:

[1]	CASALIS, M. 1991. Les familles exponentielles a variance quadratique homogene sont des lois de Wishart sur un cone sy metrique. C. R. Acad. Sci. Paris Ser. I Math. 312 537 540. \' Ź. · Zbl 0745.62051
[2]	CASALIS, M. 1992a. Un estimateur de la variance pour une famille exponentielle naturelle a fonction-variance quadratique. C. R. Acad. Sci. Paris Ser. I Math. 314 143 146. Ź. 2 Z. · Zbl 0743.62041
[3]	CASALIS, M. 1992b. Les familles exponentielles sur de fonction-variance V m am m Z. B m C. C. R. Acad. Sci. Paris Ser. I Math. 314 635 638. Ź. · Zbl 0742.62009
[4]	CONSONNI, G. and VERONESE, P. 1992. Conjugate priors for exponential families having quadratic variance functions. J. Amer. Statist. Assoc. 87 1123 1127. Z. JSTOR: · Zbl 0764.62027
[5]	DIACONIS, P. and YLVISAKER, D. 1979. Conjugate priors for exponential families. Ann. Statist. 7 269 281. Z. Z. · Zbl 0405.62011
[6]	Dy NKIN, E. 1950. The structure of semi-simple Lie algebras. Uspehi Mat. Nauk N.S. 2 Z. 59 127. Am. Math. Soc. Transl. 17 1950. Z.
[7]	FEINSILVER, P. 1986. Some classes of orthogonal poly nomials associated with martingales. Proc. Amer. Math. Soc. 98 298 302. Z. JSTOR: · Zbl 0615.60050
[8]	FEINSILVER, P. 1991. Orthogonal poly nomials and coherent states. Sy mmetries in Science 5 159 172. Plenum, New York. Z. GUTIERREZ-PENA, E. and SMITH, A. F. M. 1995. Conjugate parametrizations for natural expo nential families. J. Amer. Statist. Assoc. 90 1347 1356.
[9]	KOKONENDJI, C. 1993. Familles exponentielles naturelles reelles de fonction-variance en R \' ’ Q. Doctorat de l’Univ. Paul Sabatier, Toulouse. Z.
[10]	KOKONENDJI, C. and SESHADRI, V. 1996. On the determinant of the second derivative of the Laplace transform. Ann. Statist. 24 1813 1827. Z. d · Zbl 0868.62047
[11]	LETAC, G. 1989. Le probleme de la classification des familles exponentielles naturelles de ay ant une fonction-variance quadratique. Probability Measures on Groups IX. Lecture Notes in Math. 1306 194 215. Springer, Berlin. Z. · Zbl 0679.62010
[12]	LETAC, G. 1992. Lectures on natural exponential families and their variance-functions. Monograf. Mat. 5. Z. · Zbl 0983.62501
[13]	LETAC, G. and MORA, M. 1990. Natural real exponential families with cubic variance-functions. Ann. Statist. 18 1 37. Z. · Zbl 0714.62010
[14]	MEIXNER, J. 1934. Orthogonal Poly nomsy steme mit einer besonderen Gestalt der erzengenden Function. J. London Math. Soc. 9 6 13. Z.
[15]	MORA, M. 1986. Familles exponentielles et fonctions-variance. These de 3eme cy cle de l’Univ. Paul Sabatier, Toulouse. Z.
[16]	MORRIS, C. N. 1982. Natural exponential families with quadratic variance-function. Ann. Statist. 10 65 80. Z. · Zbl 0498.62015
[17]	POMMERET, D. 1995. Poly nomes orthogonaux asocies aux familles exponentielles naturelles. \' These de l’Univ. Paul Sabatier, Toulouse. Z.
[18]	RATNAPARKHI, M. V. 1985. Multinomial distributions. In Ency clopedia of Statistical Sciences 5 659 665. Wiley, New York. Z.
[19]	SHANBHAG, D. N. 1979. Diagonality of the Bhattachary ya matrix as a characterization. Theory Probab. Appl. 24 430 433. · Zbl 0448.60014