BIAS-ROBUST ESTIMATES OF REGRESSION-BASED ON PROJECTIONS

A new class of bias-robust estimates of multiple regression is introdu ced. If y and x are two real random variables, let T(y, x) be a univar iate robust estimate of regression of y on x through the origin. The r egression estimate T(y, x) of a random variable y on a random vector x = (x1,...,x(p))' is defined as the vector t is-an-element-of R(p) whi ch minimizes sup\\lambda\\=1\T(y - t'x,lambda'x)\s(lambda'x), where s is a robust estimate of scale. These estimates, which are called proje ction estimates, are regression, affine and scale equivariant. When th e univariate regression estimate is T(y, x) = median(y/x), the resulti ng projection estimate is highly bias-robust. In fact, we find an uppe r bound for its maximum bias in a contamination neighborhood, which is approximately twice the minimum possible value of this maximum bias f or any regression and affine equivariant estimate. The maximum bias of this estimate in a contamination neighborhood compares favorably with those of Rousseeuw's least median squares estimate and of the most bi as-robust GM-estimate. A modification of this projection estimate, who se maximum bias for a multivariate normal with mass-point contaminatio n is very close to the minimax bound, is also given. Projection estima tes are shown to have a rate of consistency of n1/2. A computational v ersion of these estimates, based on subsampling, is given. A simulatio n study shows that its small sample properties compare very favorably to those of other robust regression estimates.