Sankhya: The Indian Journal of Statistics

2005, Volume 67, Pt. 4, 699--714

Distance of a mixture from its parent distribution

Denys Pommeret, CREST-ENSAI, France

SUMMARY. In this paper we consider mixtures of distributions from a natural exponential family. Using an adequate basis of polynomials we obtain an expression for the difference between the mixed density and its parent. This technique is applied to evaluate the distance of the mixture from the parent distribution in $L^2$-norm. Bounds are also derived in $L^1$-norm and for the difference between distribution functions. We illustrate the results by some examples through gamma, Poisson and normal mixtures. Finally, more general refined  approximations of the mixture density are proposed and illustrated through a Poisson mixture model.

AMS (2000) subject classification. 62E17.

Key words and phrases. $L^1$ and $L^2$-norm, mixed gamma distribution, mixed Poisson distribution, mixed normal distribution, orthogonal polynomials.

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