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ROBUST ESTIMATION OF *k*-COMPONENT UNIVARIATE

NORMAL MIXTURES

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B. R. CLARKE^{1} AND C. R. H EATHCOTE^{2}

^{1} *School of Mathematical and Physical Sciences, Murdoch
Western Australia 6150, Australia*

^{2} *Department of Statistics, Australian National
University,*

GPO Box 4, Canberra ACT 2601, Australia
(Received June 25, 1992; revised March 25, 1993)

**Abstract.**
The estimating equations derived from minimising a
*L*_{2} distance between the empirical distribution function and the
parametric distribution representing a mixture of *k* normal
distributions with possibly different means and/or different dispersion
parameters are given explicitly. The equations are of the *M* estimator
form in which the *psi* function is smooth, bounded and has bounded
partial derivatives. As a consequence it is shown that there is a
solution of the equations which is robust. In particular there exists a
weakly continuous, Fréchet differentiable root and hence there is a
consistent root of the equations which is asymptotically normal. These
estimating equations offer a robust alternative to the maximum likelihood
equations, which are known to yield nonrobust estimators.

*Key words and phrases*:
Influence function, weak continuity,
mixtures of normals, Fréchet differentiability, consistency, asymptotic
normality, selection
functional, minimum distance estimator.

**Source**
( TeX ,
DVI ,
PS )