The Maximum Bias of Robust Covariances

Abstract

This paper deals with the maximum asymptotic bias of tiro classes of robust estimates of the dispersion matrix V of a p-dimensional random vector z, under a contamination model of the form P = (1—ε)Po+δ(x0), where P is the distribution of z, Po is a spherical distribution, and δ(x0) is a point mass at z0. Estimators VQ,α of the first class minimise the α quantile of x´V-1z among all symmetric positive-definite matrices V for some α ϵ (0,1). The "maximum volume ellipsoid" estimator proposed by Rouseauw belongs to this class with α = 0.5. These estimators have breakdown point min(α, 1 - α) for all p. The second class of estimators constat of the M-estimaton, from which the seemingly most robust member was choses; namely the Tyler estimate defined as the solution VT of Ez´VT-1z/z´z = VT. This estimator has breakdown point 1/p. The numerical results show that except for ε very close to 1/p, VT has in general a smaller máximum bias than VQ,α; and that the maximum bias of the latter may be extremely large even for a much smaller than its breakdown point.