Bias-robust estimators of multivariate scatter based on projections

Equivariant estimation of the multivariate scatter of a random vector X can be derived from a criterion of (lack of) spherical symmetry g(X). The scatter matrix is V = (ATA)-1, where A is the transformation matrix which makes AX as spherical as possible, that is, which minimizes g(AX). The new class of projection estimators is based on making the spread of univariate projections as constant as possible by choosing g(X) = supu = 1 s(uTX) -1, where s is any robust scale functional. The breakdown point of such an estimator is at least that of s, independently of the dimension p of X. In order to study the bias, we calculate condition numbers based on asymptotics and on simulations of finite samples for a spherically symmetric X, contaminated by a point mass, with the median absolute deviation as the scale measure. The simulations are done for an algorithm which is designed to approximate the projection estimator. The bias is much lower than the one of Rousseeuw's MVE-estimator, and compares favorably in most cases with two M-estimators.