ROBUST ESTIMATION OF k-COMPONENT UNIVARIATE NORMAL MIXTURES

B. R. CLARKE^{1} AND C. R. HEATHCOTE^{2}

1School of Mathematical and Physical Sciences, Murdoch University, Western Australia 6150, Australia 2Department 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 ψ 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.