Natural Exponential Families and Umbral Calculus

  • A. di Bucchianico
  • D. E. Loeb
Chapter
Part of the Progress in Mathematics book series (PM, volume 161)

Abstract

We use the Umbral Calculus to investigate the relation between natural exponential families and Sheffer polynomials. As a corollary, we obtain a new transparent proof of Feinsilver’s theorem which says that natural exponential families have a quadratic variance function if and only if their associated Sheffer polynomials are orthogonal.

Keywords

natural exponential family variance function umbral calculus Sheffer polynomials orthogonal polynomials approximation operators 
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References

  1. [1]
    Di Bucchianico, D.E. Loeb, and G.-C. Rota, Umbral calculus in Hilbert space, to appear in Rota Festschrift.Google Scholar
  2. [2]
    W.A. Al-Salam, On a characterization of Meixner’s polynomials, Quart J. Math. Oxford 2 17 (1966), 7–10.MathSciNetCrossRefGoogle Scholar
  3. [3]
    S.K. Bar-Lev, D. Bshouty, P. Enis, G. Letac, I.-L. Lu, D. Richards, The diagonal multivariate natural exponential families and their classification. J. Theoret. Probab. 7 (1994), 883–929.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, 1978.MATHGoogle Scholar
  5. [5]
    Y. Chikuse, Multivariate Meixner classes of invariant distributions, Lin. Alg. Appl. 82 (1986), 177–200.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    A.J. Dobson, An introduction to generalized linear models, Chapman and Hall, 1994.Google Scholar
  7. [7]
    P.J. Feinsilver, Special Functions Probability Semigroups, and Hamiltonian Flows, Lect. Notes in Math.696 Springer, Berlin, 1978.Google Scholar
  8. [8]
    J.M. Freeman, Orthogonality via transforms, Stud. Appl. Math. 77 (1987), 119–127.MathSciNetMATHGoogle Scholar
  9. [9]
    M.E.H. Ismail, Polynomials of binomial type and approximation theory, J. Approx. Theor. 23 (1978), 177–186.MATHCrossRefGoogle Scholar
  10. [10]
    M.E.H. Ismail and C.P. May, On a family of approximation operators, J. Math. Anal. Appl. 63 (1978), 446–462.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    S.A. Joni, Multivariate exponential operators, Stud. Appl. Math.62 (1980), 175–182.MathSciNetMATHGoogle Scholar
  12. [12]
    A.N. Kholodov, The umbral calculus and orthogonal polynomials, Acta Appl. Math. 19 (1990), 55–76.MathSciNetMATHGoogle Scholar
  13. [13]
    H.O. Lancaster, Joint probability distributions in the Meixner classes, J. Roy. Stat. Soc. B 37 (1975), 434–443.MathSciNetMATHGoogle Scholar
  14. [14]
    G. Letac, Le problème de la classification des familles exponentielles naturelles de Rd ayant une fonction variance quadratique, in: Probability measures on groups IX, Lect. Notes in Math. 1379, H. Heyer (ed.), Springer, Berlin, 1989, 192–216.CrossRefGoogle Scholar
  15. [15]
    G. Letac and M. Mora, Natural real exponential families with cubic variance functions, Ann. Stat.18 (1990), 1–37.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    C.P. May, Saturation and inverse theorems for combinations of a class of exponential-type operators, Can. J. Math.28 (1976), 1224–1250.MATHCrossRefGoogle Scholar
  17. [17]
    J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Punktion, J. London Math. Soc.9 (1934), 6–13.CrossRefGoogle Scholar
  18. [18]
    C.N. Morris, Natural exponential families with quadratic variance functions,Ann. Stat. 10 (1982), 65–80.MATHCrossRefGoogle Scholar
  19. [19]
    C.N. Morris, Natural exponential families with quadratic variance functions: statistical theory, Ann. Stat.11 (1983), 515–529.MATHCrossRefGoogle Scholar
  20. [20]
    D. Pommeret, Familles exponentielles naturelles et algèbres de Lie,Exposition. Math. 14 (1996), 353–381.MathSciNetMATHGoogle Scholar
  21. [21]
    D. Pommeret, Orthogonal polynomials and natural exponential families, Test 5 (1996), 77–111.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    G.-C. Rota, D. Kahaner, and A. Odlyzko, On the foundations of combinatorial theory VII. Finite operator calculus,J. Math. Anal. Appl. 42 (1973), 684–760.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    S.M. Roman, The Umbral Calculus, Academic Press, 1984.Google Scholar
  24. [24]
    S.M. Roman, P. De Land, R. Shiflett, and H. Schultz, The umbral calculus and the solution to certain recurrence relations, J. Comb. Inf. Sys. Sci. 8 (1983), 235–240.MATHGoogle Scholar
  25. [25]
    S.D. Silvey, Statistical inference, Monographs on Statistics and Applied Probability 7, Chapman and Hall, 1991.Google Scholar
  26. [26]
    I.M. Sheffer, Some properties of polynomials of type zero, Duke Math. J.5 (1939), 590–622.MathSciNetCrossRefGoogle Scholar
  27. [27]
    X.-H. Sun, New characteristics of some polynomial sequences in combinatorial theory,J. Math. Anal. Appl. 175 (1993), 199–205.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser 1998

Authors and Affiliations

  • A. di Bucchianico
    • 1
  • D. E. Loeb
    • 2
  1. 1.Dept. of Mathematics and Computing ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Laboratoire Bordelais de Recherche en InformatiqueUniversité de Bordeaux ITalence CedexFrance

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