**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.