Bias-Robust Estimates of Regression Based on Projections

Abstract

A new class of bias-robust estimates of multiple regression is introduced. If $y$ and $x$ are two real random variables, let $T(y, x)$ be a univariate robust estimate of regression of $y$ on $x$ through the origin. The regression estimate $\mathbf{T}(y, \mathbf{x})$ of a random variable $y$ on a random vector $\mathbf{x} = (x_1,\cdots, x_p)'$ is defined as the vector $\mathbf{t} \in \mathfrak{R}^p$ which minimizes $\sup_{\|\mathbf{\lambda}\| = 1} \mid T(y - \mathbf{t'x, \lambda' x}) \mid s(\mathbf{\lambda'x})$, where $s$ is a robust estimate of scale. These estimates, which are called projection estimates, are regression, affine and scale equivariant. When the univariate regression estimate is $T(y, x) =$ median $(y/x)$, the resulting projection estimate is highly bias-robust. In fact, we find an upper bound for its maximum bias in a contamination neighborhood, which is approximately twice the minimum possible value of this maximum bias for 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 bias-robust GM-estimate. A modification of this projection estimate, whose maximum bias for a multivariate normal with mass-point contamination is very close to the minimax bound, is also given. Projection estimates are shown to have a rate of consistency of $n^{1/2}$. A computational version of these estimates, based on subsampling, is given. A simulation study shows that its small sample properties compare very favorably to those of other robust regression estimates.