The maximum bias of robust covariances

This paper deals with the maximum asymptotic bias of two classes of robust estimates of the dispersion matrix V of a p-dimensional random vector x, under a contamination model of the form , where P is the distribution of x,P 0 is a spherical distribution, and δ(x 0) is a point mass at x 0. Estimators VQ,α of the first class minimize the a quantile of x'V-1x among all symmetric positive-definite matrices V for some α∊(0,1). The “mimimum volume ellipsoid” estimator proposed by Rousseeuw 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 consist of the M-estimators, from which the seemingly most robust member was chosen; namely the Tyler estimate defined as the solution VT of . This estimator has breakdown point 1/p. The numerical results show that except for ∊ very close to 1/p VT has in general a smaller maximum bias than VQ,α and that the maximum bias of the latter may be extremely large even for e much smaller than its breakdown point.