Polynomial expansions of density of power mixtures

Pommeret, Denys

ESAIM: Probability and Statistics, Tome 11 (2007), pp. 248-263.

For any given random variable $Y$ with infinitely divisible distribution in a quadratic natural exponential family we obtain a polynomial expansion of the power mixture density of $Y$. We approach the problem generally, and then consider certain distributions in greater detail. Various applications are indicated and the results are also applied to obtain approximations and their error bounds. Estimation of density and goodness-of-fit test are derived.

[1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions. Dover, New York (1972).

[2] O.E. Barndorff-Nielsen, Information and Exponential Families. Wiley, New York (1978).

[3] M. Casalis, The $2d+4$ simple quadratic natural exponential families on ${ℝ}^d$. Ann. Statist. 24 (1996) 1828-1854. | Zbl

[4] P. Feinsilver, Some classes of orthogonal polynomials associated with martingales. Proc. A.M.S. 98 (1986) 298-302. | Zbl

[5] W. Feller, An Introduction to Probability Theory and Its Applications. Vol. I, Wiley (1966a). | Zbl

[6] W. Feller, An Introduction to Probability Theory and Its Applications. Vol. II, Wiley (1966b). | MR | Zbl

[7] Y. Fujikoshi and R. Shimizu, Asymptotic expansions for univariate and multivariate distributions. J. Multivariate Anal. 30 (1989) 279-291. | Zbl

[8] P. Hall, Polynomial Expansion of Density and Distribution Functions of Scale Mixtures. J. Multivariate Anal. 11 (1981) 173-184. | Zbl

[9] B. Jorgensen, The Theory of Dispersion models. Chapman & Hall, London (1997). | MR | Zbl

[10] J. Keilson and F.W. Steutel, Mixtures of distributions, moment inequalities and measures of exponentiality and normality. Ann. Probab. 2 (1974) 112-130. | Zbl

[11] R. Koekoek and R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Report no. 94-05, Delft University of Technology, Faculty of Technical Mathematics and Informatics (1994).

[12] G. Letac, Lectures on natural exponential families and their variance functions. Instituto de matemática pura e aplicada: Monografias de matemática 50, Río de Janeiro, Brésil (1992). | MR | Zbl

[13] J. Meixner, Orthogonal Polynomsysteme mit einer besonderen Gestalt der erzengenden Function, J. London Math. Soc. 9 (1934) 6-13. | Zbl

[14] C.N. Morris, Natural exponential families with quadratic variance functions. Ann. Statist. 10 (1982) 65-82. | Zbl

[15] D. Pommeret, Orthogonal polynomials and natural exponential families. Test 5 (1996) 77-111. | Zbl

[16] D. Pommeret, Multidimensional Bhattacharyya Matrices and Exponential Families. J. Multivariate Anal. 63 (1997) 105-118. | Zbl

[17] J.C.W. Rayner and D.J. Best, Smooth Tests of Goodness of Fit. Oxford University Press, New York (1989). | MR | Zbl

[18] R.F. Serfozo, Random Time Transformations of Semi-Markov Processes. Ann. Math. Statist. 42 (1971) 176-188. | Zbl

[19] R. Shimizu, Error bounds for asymptotic expansion of the scale mixture of the normal distribution. Ann. Inst. Statist. Math. 39 (1987) 611-622. | Zbl

[20] R. Shimizu, Expansion of the Scale Mixture of the Multivariate Normal Distributions with Error Bound Evaluated in the ${\mathrm{L}}^1$-Norm. J. Multivariate Anal. 5 (1995) 126-138. | Zbl

[21] R. Shimizu and Y. Fujikoshi, Sharp error bound for asymptotic expansions of distribution functions for scale mixture. Ann. Inst. Statist. Math. 49 (1997) 285-297. | Zbl