Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T04:04:27.897Z Has data issue: false hasContentIssue false
  • English
  • Français

Polynomial expansions of density of power mixtures

Published online by Cambridge University Press:  19 June 2007

Denys Pommeret
Denys Pommeret*
Affiliation:
CREST - ENSAI, Rue Blaise Pascal, BP 37203, 35172 Bruz Cedex, France; [email protected]
Get access

Abstract

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

Keywords

Approximationconvolutionerror boundgoodness-of-fit testmixed distributionorthogonal polynomialsscale mixture.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 11: Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthdaySpecial Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday , June 2007 , pp. 248 - 263
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions. Dover, New York (1972).
O.E. Barndorff-Nielsen, Information and Exponential Families. Wiley, New York (1978).
Casalis, M., The 2d + 4 simple quadratic natural exponential families on $\xR^d$ . Ann. Statist. 24 (1996) 1828–1854.
Feinsilver, P., Some classes of orthogonal polynomials associated with martingales. Proc. A.M.S. 98 (1986) 298–302. CrossRef
W. Feller, An Introduction to Probability Theory and Its Applications. Vol. I, Wiley (1966a).
W. Feller, An Introduction to Probability Theory and Its Applications. Vol. II, Wiley (1966b).
Fujikoshi, Y. and Shimizu, R., Asymptotic expansions for univariate and multivariate distributions. J. Multivariate Anal. 30 (1989) 279–291. CrossRef
Hall, P., Polynomial Expansion of Density and Distribution Functions of Scale Mixtures. J. Multivariate Anal. 11 (1981) 173–184. CrossRef
B. Jorgensen, The Theory of Dispersion models. Chapman & Hall, London (1997).
J. Keilson and F.W. Steutel, Mixtures of distributions, moment inequalities and measures of exponentiality and normality. Ann. Probab. 2 (1974) 112–130.
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).
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).
J. Meixner, Orthogonal Polynomsysteme mit einer besonderen Gestalt der erzengenden Function, J. London Math. Soc. 9 (1934) 6–13.
Morris, C.N., Natural exponential families with quadratic variance functions. Ann. Statist. 10 (1982) 65–82. CrossRef
Pommeret, D., Orthogonal polynomials and natural exponential families. Test 5 (1996) 77–111. CrossRef
Pommeret, D., Multidimensional Bhattacharyya Matrices and Exponential Families. J. Multivariate Anal. 63 (1997) 105–118. CrossRef
J.C.W. Rayner and D.J. Best, Smooth Tests of Goodness of Fit. Oxford University Press, New York (1989).
Serfozo, R.F., Random Time Transformations of Semi-Markov Processes. Ann. Math. Statist. 42 (1971) 176–188. CrossRef
Shimizu, R., Error bounds for asymptotic expansion of the scale mixture of the normal distribution. Ann. Inst. Statist. Math. 39 (1987) 611–622. CrossRef
Shimizu, R., Expansion of the Scale Mixture of the Multivariate Normal Distributions with Error Bound Evaluated in the L1-Norm. J. Multivariate Anal. 5 (1995) 126–138. CrossRef
Shimizu, R. and Fujikoshi, Y., Sharp error bound for asymptotic expansions of distribution functions for scale mixture. Ann. Inst. Statist. Math. 49 (1997) 285–297. CrossRef