A Distribution-Free $M$-Estimator of Multivariate Scatter

Abstract

The existence and uniqueness of a limiting form of a Huber-type $M$-estimator of multivariate scatter is established under certain conditions on the observed sample. These conditions hold with probability one when sampling randomly from a continuous multivariate distribution. The existence of the estimator is proven by showing that it is the limiting point of a specific algorithm. Hence, the proof is constructive. For continuous populations, the estimator of multivariate scatter is shown to be strongly consistent and asymptotically normal. An important property of the estimator is that its asymptotic distribution is distribution-free with respect to the class of continuous elliptically distributed populations. This distribution-free property also holds for the finite sample size distribution when the location parameter is known. In addition, the estimator is the "most robust" estimator of the scatter matrix of an elliptical distribution in the sense of minimizing the maximum asymptotic variance.