Central limit theorem and inflence function for the MCD estimators at general multivariate distributions

Abstract

The minimum covariance determinant (MCD) estimators of multivariate location and scatter are robust alternatives to the ordinary sample mean and sample covariance matrix. Nowadays they are used to determine robust Mahalanobis distances in a reweighting procedure, and used as robust plug-ins in all sorts of multivariate statistical techniques which need a location and/or covariance estimate, such as principal component analysis, factor analysis, discriminant analysis and linear multivariate regression. For this reason, the distributional and the robustness properties of the MCD estimators are essential for conducting inference and perform robust estimation in several statistical models. Butler, Davies and Jhun (1993) show asymptotic normality only for the MCD location estimator, whereas the MCD covariance estimator is only shown to be consistent. Croux and Haesbroeck (1999) give the expression for the influence function of the MCD covariance functional and use this to compute limiting variances of the MCD covariance estimator. However, the expression is obtained under the assumption of existence, continuity and differentiability of the MCD-functionals at perturbed distributions, which is not proven. Moreover, the computation of the limiting variances relies on the von Mises expansion of the estimator, which has not been established. In this presentation we define the MCD functional by means of trimming functions which are in a wide class of measurable functions. The class is very flexible and allows a uniform treatment at general probability measures, including empirical measures and perturbed measures. We prove existence of the MCD functional for any multivariate distribution P and provide a separating ellipsoid property for the functional. Furthermore, we prove continuity of the functional, which also yields strong consistency of the MCD estimators. Finally, we derive an asymptotic expansion of the functional, from which we rigorously derive the influence function, and establish a central limit theorem for both MCD-estimators. All results are obtained under very mild conditions on P and essentially all conditions are automatically satisfied for distributions with a density. For distributions with an elliptically contoured density that is unimodal we do not need any extra condition and one may recover the results in Butler, Davies and Jhun (1993) and Croux