Abstract
A natural exponential family (NEF)F in ℝn,n>1, is said to be diagonal if there existn functions,a1,...,an, on some intervals of ℝ, such that the covariance matrixVF(m) ofF has diagonal (a1(m1),...,an(mn)), for allm=(m1,...,mn) in the mean domain ofF. The familyF is also said to be irreducible if it is not the product of two independent NEFs in ℝk and ℝn-k, for somek=1,...,n−1. This paper shows that there are only six types of irreducible diagonal NEFs in ℝn, that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: IfF is an NEF in ℝn, under what conditions is its projectionp(F) in ℝk, underp(x1,...,xn)∶=(x1,...,xk),k=1,...,n−1, still an NEF in ℝk? The answer turns out to be rather predictable. It is the case if, and only if, the principalk×k submatrix ofVF(m1,...,mn) does not depend on (mk+1,...,mn).
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Askey, R. (1974). Certain rational functions whose power series have positive coefficients, II.Siam J. Math. Anal. 5, 53–57.
Askey, R., and Gasper, G. (1972). Certain rational functions whose power series have positive coefficients.Amer. Math. Monthly 79, 327–341.
Askey, R., and Gasper, G. (1977). Convolution structures for Laguerre polynomials.Journal d'Analyse Math. 31, 48–68.
Auerhan, J., and Letac, G. (1991). Which negative multinomial distributions are infinitely divisible? Preprint.
Bar-Lev, S. K., Bshouty, D., and Enis, P. (1991). Multivariate natural exponential families with Poisson marginals. Technical Report 91-2, Department of Statistics, State University of New York at Buffalo.
Barndorff-Nielsen, O. (1978).Information and Exponential Families in Statistical Theory, Wiley, New York.
Barndorff-Nielsen, O. (1980). Conditionality resolutions.Biometrika 67, 293–310.
Barndorff-Nielsen, O., Blaesild, P., and Seshadri, V. (1992). Multivariate distributions with generalized inverse Gaussian marginals and associated mixtures.Canadian J. Statist. 20, 109–120.
Barndorff-Nielsen, O., and Koudou, A. (1992). Cuts in natural exponential families. Preprint.
Brown, L. D. (1986).Fundamentals of Statistical Exponential Families, IMS, Hayward, California.
Bshouty, D., and Letac, G. (1990). The projection of a natural exponential family and the bivariate Morris families. Université Paul-Sabatier, Preprint.
Buja, A. (1990). Remarks on functional canonical variates, alternating least squares methods and ACE.Ann. Statist. 18, 1032–1069.
Casalis, M. (1990). Families exponentielles naturelles invariantes par un sous-groupe affine. Thèse, Université Paul-Sabatier, Toulouse.
Casalis, M. (1991). Les familles exponentielles à variance quadratique homogène sont des lois de Wishart sur un cone symétrique.C. R. Acad. Sci. Paris Sér. I 312, 537–540.
Casalis, M. (1992). Les familles exponentielle sur ℝ2 de fonction varianceV(m)=am⊗m+B(m)+C.C. R. Acad. Sci. Paris Sér. I 314, 635–638.
Chentsov, N. N. (1982).Statistical Decision Rules and Optimal Inference, Amer. Math. Soc., Providence, Rhode Island.
Doss, D. C. (1979). Definition and characterization of multivariate negative binomial distribution.J. Multivariate Anal. 9, 460–464.
Eagleson, G. K. (1964). Polynomial expansions of bivariate distributions.Ann. Math. Statist. 35, 1208–1215.
Eagleson, G. K. (1969). A characterization theorem for positive definite sequences on the Krawtchouk polynomials.Austral. J. Statist. 11, 29–38.
Erdélyi, A. (1953).Higher Transcendental Functions, Vol. 1, McGraw Hill, New York.
Evans, S. N. (1991). Association and infinite divisibility for the Wishart distribution and its diagonal marginals.J. Multivariate Anal. 36, 199–203.
Gradshteyn, I. S., and Ryzhik, I. M. (1980).Tables of Integrals, Series and Products, Fourth ed. Academic, New York.
Griffiths, R. C. (1984). Characterization of infinitely divisible multivariate gamma distributions.J. Multivariate Anal. 15, 13–20.
Griffiths, R. C. (1988). Orthogonal expansions. In Johnson, N. L., and Kotz, S. (eds.),Encyclopedia of Statistical Sciences, Wiley, New York,5, 530–536.
Griffiths, R. C., and Milne, R. K. (1987). A class of infinitely divisible multivariate negative binomial distributions.J. Multivariate Anal. 22, 13–23.
Griffiths, R. C., Milne, R. K., and Wood, R. (1979). Aspects of correlation in bivariate Poisson distributions and processes.Austral. J. Statist. 21, 238–255.
Hlanek, J. (1969).Transformées de Laplace des Fonctions de Plusieurs Variables, Eyrolles, Paris.
Jørgensen, B. (1987). Exponential dispersion models (with discussion).J. Roy. Statist. Soc. Ser. B 49, 127–162.
Kocherlakota, S. (1989). A note on the bivariate binomial distribution.Statist. Prob. Lett. 8, 21–24.
Lai, C. D., and Vere-Jones, D. (1979). Odd man out-the Meixner hypergeometric distribution.Austral. J. Statist. 21, 256–265.
Lancaster, H. O. (1969).The Chi-Squared Distribution, Wiley, New York.
Lancaster, H. O. (1975). Joint probability distributions in the Meixner classes.J. Roy. Statist. Soc. Ser. B 37, 434–443.
Letac, G. (1989). A characterization of the Wishart exponential families by an invariance property.J. Th. Prob. 2, 71–86.
Letac, G. (1989). Le problème de la classification des familles exponentielles naturalles dans ℝd de fonction variance quadratique. Springer, Berlin.Probability on Groups IX.Lecture Notes in Math.1306, 194–215.
Letac, G., and Mora, M. (1990). Natural real exponential families with cubic variance functions.Ann. Statist. 18, 1–37.
Lu, I. and Richards, D. (1991). Multivariate natural exponential families with quadratic variance function. University of Virginia, Preprint.
Lukacs, E. (1970).Characteristic Functions, Second ed. Griffin, London.
Morris, C. N. (1982). Natural exponential families with quadratic variance functions.Ann. Statist. 10, 65–80.
Muirhead, R. J. (1982).Aspects of Multivariate Statistical Theory, Wiley, New York.
Rainville, E. D. (1960).Special Functions, Macmillan, New York.
Ratnaparkhi, M. V. (1988). Multinomial distributions. In Johnson, N. L. and Kotz, S. (eds.). InEncyclopedia of Statistical Sciences, Wiley, New York,7, 659–665.
Sarmanov, O. V., and Bratoeva, Z. N. (1967). Probabilistic properties of bilinear expansions of Hermite polynomials.Theory Prob. Appl. 12, 470–481.
Seshadri, V. (1987). Contribution to discussion of paper by B. Jørgensen: Exponential dispersion models.J. Roy. Statist. Soc. Ser. B 49, 156.
Szegö, G. (1975).Orthogonal Polynomials, Fourth ed. Amer. Math. Soc., Providence, Rhode Island.
Teicher, H. (1954). On the multivariate Poisson distribution.Skand. Aktuarietids. 37, 1–9.
Tyan, S., Derin, H., and Thomas, J. B. (1976). Two necessary conditions on the representation of bivariate distributions by polynomials.Ann. Statist. 4, 216–222.
Tyan, S., and Thomas, J. B. (1975). Characterization of a class of bivariate distribution functions.J. Multivariate Anal. 5, 227–235.
Watson, G. N. (1952).A Treatise on the Theory of Bessel Functions, Second ed. Cambridge University Press, Cambridge.
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Bar-Lev, S.K., Bshouty, D., Enis, P. et al. The diagonal multivariate natural exponential families and their classification. J Theor Probab 7, 883–929 (1994). https://doi.org/10.1007/BF02214378
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DOI: https://doi.org/10.1007/BF02214378