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Exponential families with variance functions in\(\sqrt {\Delta P} (\sqrt \Delta )\): Seshadri’s class: Seshadri’s class

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This paper presents a classification of the setS 3 of all natural exponential families (NEF) on ℝ which have a variance function of the form\(\sqrt {\Delta P} (\sqrt \Delta )\), whereP is a polynomial of degree 3 and Δ is an affine function of the mean of the NEF. Particular cases have been considered previosly by V. Seshadri and can be obtained by a Lindsay transform of the NEF with cubic variance, as classified by Marianne Mora.S 3 may be split into six types and we provide a probabilistic interpretation of each of them; in particular, we show that the literature on random mappings provides several examples of discrete elements ofS 3. The final result gives the closure ofS 3 under the topology of weak convergence.

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References

  • Bar-Lev, S. K. and Enis, P. (1987). Reproducibility and natural exponential families with power variance functions.Ann. Statist. 14, 1507–1522.

    MathSciNet  Google Scholar 

  • Barndorff-Nielsen, O. E. (1978).Information and Exponential Families in Statistical Theory. Chichester: Ellis Horwood.

    MATH  Google Scholar 

  • Berg, S. and Nowicki, K. (1991). Statistical inference for a class of modified power series distributions with applications to random mapping theory.J. Statist. Planning and Inference 28, 247–261.

    Article  MathSciNet  Google Scholar 

  • Consul, P. C. (1989).Generalized Poisson Distributions: Properties and Applications. New York: Marcel Dekker.

    MATH  Google Scholar 

  • Dieudonné, J. (1971).Infinitesimal Calculus. Boston: Houghton Mifflin

    Google Scholar 

  • Dudley, R. M. (1989).Real Analysis and Probability. California: Wadsworth and Brooks Cole.

    MATH  Google Scholar 

  • Harris, T. E. (1963).The Theory of Branching Processes Berlin: Springer-Verlag.

    MATH  Google Scholar 

  • Janardan, K. G. and Rao, B. R. (1983). Lagrange distributions of the second kind and weighted distributions.J. Appl. Math. 43, 302–313.

    MathSciNet  Google Scholar 

  • Jørgensen, B. (1987). Exponential dispersion models.J. Roy. Statist. Soc. B 49, 127–162.

    Google Scholar 

  • Jørgensen, B., Martínez, J. R. and Tsao, M. (1994). Asymptotic behaviour of the variance function.Scandinavian J. Statist. 21, 223–243.

    Google Scholar 

  • Kokonendji, C. C. (1992). Caractérisation de fonctions variance de Seshadri des familles exponentielles sur.R. C. R. Acad. Sci. Paris, Série I. 314, 1063–1068.

    MathSciNet  Google Scholar 

  • Kokonendji, C. C. and Seshadri, V. (1992). La méthode de Lindsay appliquée à la construction de familles exponentielles de fonctions variance de degré 4 en\(\sqrt m \).C. R. Acad. Sci. Paris, Série I. 314, 305–308.

    MathSciNet  Google Scholar 

  • Kokonendji, C. C. and Seshadri V. (1994). The Lindsay transform of natural exponential families.Canadian J. Statist. 22, 259–272.

    MathSciNet  Google Scholar 

  • Letac, G. (1986). La réciprocité des familles exponentielles naturelles surℝ. C. R. Acad. Sci. Paris, Série I. 303, 61–64.

    MathSciNet  Google Scholar 

  • Letac, G. (1991). The classification of natural exponential families by their variance functions.Proceedings of the 48th session of the International Statistical Institute.54, #3.

  • Letac, G. (1992).Lectures on Natural Exponential Families and their Variance Functions. Rio de Janeiro: I.M.P.A.

    MATH  Google Scholar 

  • Letac, G. and Mora, M. (1986). Sur les fonctions-variance des familles exponentielles naturelles surℝ. C. R. Acad. Sci. Paris, Série I. 302, 551–554.

    MathSciNet  Google Scholar 

  • Letac, G. and Mora, M. (1990). Natural real exponential families with cubic variance functions.Ann. Statist. 18, 1–37.

    MathSciNet  Google Scholar 

  • Loève, M. (1963).Probability Theory. Toronto, New York, London: Van Norstrand.

    MATH  Google Scholar 

  • Mora, M. (1986). Classification des fonctions-variance cubiques des familles exponentielles surℝ. C. R. Acad. Sci. Paris, Série I. 302, 587–590.

    MathSciNet  Google Scholar 

  • Mora, M. (1990). La convergence des fonctions variance des familles exponentielles naturelles.Annales de la Faculté des Sci. de Toulouse, Série 5, XI fasc. 2, 105–120.

    MathSciNet  Google Scholar 

  • Morris, C. N. (1982). Natural exponential families with quadratic variance functions.Ann. Statist. 10, 65–80.

    MathSciNet  Google Scholar 

  • Rainville, E. D. (1960).Special Functions. New York: Chelsea.

    MATH  Google Scholar 

  • Seshadri, V. (1991). Finite mixtures of natural exponential families.Canadian J. Statist. 19, 437–445.

    MathSciNet  Google Scholar 

  • Wendel, J. G. (1975). Left continuous random walk and Lagrange expansion.Amer. Math. Monthly. 82, 494–499.

    Article  MathSciNet  Google Scholar 

  • Yanagimoto, T. (1989). The inverse binomial distribution as a statistical model.Comm. Statist. Theory and Methods 18, 3625–3633.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Statistique et Probabilités, Université Paul Sabatier-CNRS U.R.A. D0745, 118, Route de Narbonne, 31062, Toulouse cedex, France

    Célestin C. Kokonendji

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  1. Célestin C. Kokonendji
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Kokonendji, C.C. Exponential families with variance functions in\(\sqrt {\Delta P} (\sqrt \Delta )\): Seshadri’s class: Seshadri’s class. Test 3, 123–172 (1994). https://doi.org/10.1007/BF02562698

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  • DOI: https://doi.org/10.1007/BF02562698

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