# Polynomial expansions of density of power mixtures

ESAIM: Probability and Statistics (2007)

- Volume: 11, page 248-263
- ISSN: 1292-8100

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topPommeret, Denys. "Polynomial expansions of density of power mixtures." ESAIM: Probability and Statistics 11 (2007): 248-263. <http://eudml.org/doc/250090>.

@article{Pommeret2007,

abstract = {
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.
},

author = {Pommeret, Denys},

journal = {ESAIM: Probability and Statistics},

keywords = {Approximation; convolution; error bound;
goodness-of-fit test; mixed distribution; orthogonal polynomials;
scale mixture.; approximation; goodness-of-fit test; scale mixture},

language = {eng},

month = {6},

pages = {248-263},

publisher = {EDP Sciences},

title = {Polynomial expansions of density of power mixtures},

url = {http://eudml.org/doc/250090},

volume = {11},

year = {2007},

}

TY - JOUR

AU - Pommeret, Denys

TI - Polynomial expansions of density of power mixtures

JO - ESAIM: Probability and Statistics

DA - 2007/6//

PB - EDP Sciences

VL - 11

SP - 248

EP - 263

AB -
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.

LA - eng

KW - Approximation; convolution; error bound;
goodness-of-fit test; mixed distribution; orthogonal polynomials;
scale mixture.; approximation; goodness-of-fit test; scale mixture

UR - http://eudml.org/doc/250090

ER -

## References

top- M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions. Dover, New York (1972). Zbl0543.33001
- O.E. Barndorff-Nielsen, Information and Exponential Families. Wiley, New York (1978). Zbl0387.62011
- M. Casalis, The 2d + 4 simple quadratic natural exponential families on ${}^{d}$. Ann. Statist.24 (1996) 1828–1854. Zbl0867.62042
- P. Feinsilver, Some classes of orthogonal polynomials associated with martingales. Proc. A.M.S.98 (1986) 298–302. Zbl0615.60050
- W. Feller, An Introduction to Probability Theory and Its Applications. Vol. I, Wiley (1966a). Zbl0138.10207
- W. Feller, An Introduction to Probability Theory and Its Applications. Vol. II, Wiley (1966b). Zbl0138.10207
- Y. Fujikoshi and R. Shimizu, Asymptotic expansions for univariate and multivariate distributions. J. Multivariate Anal.30 (1989) 279–291. Zbl0676.62020
- P. Hall, Polynomial Expansion of Density and Distribution Functions of Scale Mixtures. J. Multivariate Anal.11 (1981) 173–184. Zbl0473.62020
- B. Jorgensen, The Theory of Dispersion models. Chapman & Hall, London (1997). Zbl0928.62052
- J. Keilson and F.W. Steutel, Mixtures of distributions, moment inequalities and measures of exponentiality and normality. Ann. Probab. 2 (1974) 112–130. Zbl0325.60019
- 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. Zbl60.0293.01
- C.N. Morris, Natural exponential families with quadratic variance functions. Ann. Statist.10 (1982) 65–82. Zbl0498.62015
- D. Pommeret, Orthogonal polynomials and natural exponential families. Test5 (1996) 77–111. Zbl0960.62517
- D. Pommeret, Multidimensional Bhattacharyya Matrices and Exponential Families. J. Multivariate Anal.63 (1997) 105–118. Zbl0909.62053
- J.C.W. Rayner and D.J. Best, Smooth Tests of Goodness of Fit. Oxford University Press, New York (1989). Zbl0731.62064
- R.F. Serfozo, Random Time Transformations of Semi-Markov Processes. Ann. Math. Statist. 42 (1971) 176–188. Zbl0217.50501
- R. Shimizu, Error bounds for asymptotic expansion of the scale mixture of the normal distribution. Ann. Inst. Statist. Math.39 (1987) 611–622. Zbl0638.62017
- R. Shimizu, 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. Zbl0877.62053
- 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. Zbl0890.62010

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