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Orthogonal polynomials and natural exponential families

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Summary

There exist several different characterizations of the class of quadratic natural exponential families onR, two of which use orthogonal polynomials. In Feinsilver (1986), the polynomials result from the derivation of the probability densities while Meixner (1934) adopts an exponential generating function. In this paper, we consider multidimensional extensions of their results which still yield quadratic or simple quadratic natural exponential families.

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References

  • Casalis, M. (1990).Familles Exponentielles Naturelles Invariantes par un Groupe. Thèse de l’Université Paul Sabatier de Toulouse.

  • Casalis, M. (1992). Les familles exponentielles sur R2 de fonction-varianceam×m+B(m)+C C. R. Acad. Sc. 314, Série I, 635–638.

    MathSciNet  Google Scholar 

  • Casalis, M. (1994). Les 2d+4 types de familles exponentielles naturelles quadratiques simples sur Rd.C. R. Acad. Sc. 318, Série I, 261–264.

    MathSciNet  Google Scholar 

  • Feinsilver, P. (1986). Some classes of orthogonal polynomials associated with martingales.Proc. Amer. Math. Soc. 98, 298–302.

    Article  MathSciNet  Google Scholar 

  • Feinsilver, P. (1991). Orthogonal polynomials and coherent states.Symmetries in Science: Plenum Press, 159–192.

  • Kahaner, D., Odlykzo, A. and Rota, G. C. (1973). Finite operator calculus.J. Math. Anal. Appl. 42, 685–760.

    Google Scholar 

  • Labeye-Voisin, E. and Pommeret, D. (1995). Polynômes orthogonaux associés aux familles exponentielles naturelles sur Rd C. R. Acad. Sc. 320, Série I, 79–84.

    MathSciNet  Google Scholar 

  • Laha, R. G. and Lukas, E. (1960). On a problem connected with quadratic regression.Biometrika 47, 335–343.

    MathSciNet  Google Scholar 

  • Lancaster, H. O. (1975). Joint probability distributions in the Meixner classes.J. Roy. Statist. Soc. 37, 434–443.

    MathSciNet  Google Scholar 

  • Letac, G. (1992)Lectures on Natural Exponential Families and their Variance Functions, Rio, Brasil: Instituto de Matemática Pura e Aplicada: Monografias de matemática50.

    MATH  Google Scholar 

  • Meixner, J. (1934). Orthogonal Polynomsysteme mit einer besonderen Gestalt der erzengenden Function.J. London Math. Soc. 9, 6–13.

    Google Scholar 

  • Mora, M. (1986).Familles Exponentielles Naturelles et Fonctions Variances. Thèse de l’Université Paul Sabatier de Toulouse.

  • Morris, C. N. (1982). Natural exponential families with quadratic variance functions.Ann. Statist. 10, 65–82.

    MathSciNet  Google Scholar 

  • Tratnik, M. V. (1989). Multivariable Meixner, Krawtchouk and Meixner-Pollaczek polynomials.J. Math. Phys. 30(12), 2740–2749.

    Article  MathSciNet  Google Scholar 

  • Viennot, G. (1983).Une Théorie Combinatoire des Polynômes Orthogonaux Généraux. Notes de conférences données à l’Université du Quebec à Montréal.

  • Xu, Y. (1993). On multivariate orthogonal polynomials.SIAM J. Math. Anal. 24, 783–793.

    Article  MathSciNet  Google Scholar 

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Pommeret, D. Orthogonal polynomials and natural exponential families. Test 5, 77–111 (1996). https://doi.org/10.1007/BF02562683

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  • DOI: https://doi.org/10.1007/BF02562683

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