The Annals of Probability

Characterization of the cubic exponential families by orthogonality of polynomials

Abdelhamid Hassairi and Mohammed Zarai

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Abstract

This paper introduces a notion of 2-orthogonality for a sequence of polynomials to give extended versions of the Meixner and Feinsilver characterization results based on orthogonal polynomials. These new versions subsume the Letac–Mora characterization of the real natural exponential families having cubic variance function.

Article information

Source
Ann. Probab., Volume 32, Number 3B (2004), 2463-2476.

Dates
First available in Project Euclid: 6 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1091813620

Digital Object Identifier
doi:10.1214/009117904000000522

Mathematical Reviews number (MathSciNet)
MR2078547

Zentralblatt MATH identifier
1056.62015

Subjects
Primary: 60J15
Secondary: 60E10: Characteristic functions; other transforms

Keywords
Exponential family variance function Sheffer polynomials orthogonal polynomials

Citation

Hassairi, Abdelhamid; Zarai, Mohammed. Characterization of the cubic exponential families by orthogonality of polynomials. Ann. Probab. 32 (2004), no. 3B, 2463--2476. doi:10.1214/009117904000000522. https://projecteuclid.org/euclid.aop/1091813620


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References

  • Feinsilver, P. (1986). Some classes of orthogonal polynomials associated with martingales. Proc. Amer. Math. Soc. 98 298--302.
  • Hassairi, A. (1992). La classification des familles exponentielles naturelles sur $\R^n$ par l'action du groupe linéaire de $\R^n+1$. C. R. Acad. Sci. Paris 315 207--210.
  • Letac, G. (1992). Lectures on Natural Exponential Families and their Variance Functions. Instituto de Mathematica Pure e Aplicada, Rio de Jeneiro.
  • Letac, G. and Mora, M. (1990). Natural real exponential families with cubic variance functions. Ann. Statist. 18 1--37.
  • Meixner, J. (1934). Orthogonal Polynomsysteme mit einer besonderen Gestalt der erzengenden Function. J. London Math. 9 6--13.
  • Morris, C. N. (1982). Natural exponential families with quadratic variance function. Ann. Statist. 10 65--80.
  • Pommeret, D. (1996). Orthogonal polynomials and natural exponential families. Test 5 77--111.
  • Rainville, E. D. (1960). Special Functions. Macmillan, New York.
  • Sheffer, I. M. (1939). Some properties of polynomial sets type 0. Duke Math. J. 5 590--622.
  • Tratnik, M. V. (1989). Multivariable Meixner, Krawtchouk and Meixner--Pollaczek polynomials. J. Math. Phys. 30 2740--2749.