Min-Max Bias Robust Regression

摘要

This paper considers the problem of minimizing the maximum asymptotic bias of regression estimates over epsilon-contamination neighborhoods for the joint distribution of the response and carriers. Two classes of estimates are treated: (1) M-estimates with bounded function rho applied to the scaled residuals, using a very general class of scale estimates, and (2) Bounded influence function type generalized M-estimates. Estimates in the first class are obtained as the solution of a minimization problem, while estimates in the second class are specified by an estimating equation. The first class of M-estimates is sufficiently general to include both Huber Proposal 2 simultaneous estimates of regression coefficients and residuals scale, and Rousseeuw-Yohai S-estimates of regression. It is shown than an S-estimate based on a jump-function type rho solves the min-max bias problem for the class of M-estimates with very general scale. This estimate is obtained by the minimization of the alpha-quantile of the squared residuals, where alpha=(epsilon) depends on the fraction of contamination epsilon. When epsilon approaches limit of .5, alpha (epsilon) approaches limit of .5 and the min-max estimator approaches the least median of squared residuals estimator introduced by Rousseeuw. For the bounded influence class of GM-estimates, it is shown the a sign type nonlinearity yields the min-max estimate. This estimate coincides with the minimum gross-error sensitivity GM-estimate.