Min-max Bias Robust Regression

This the problem maximum asymptotic bias of regression estimates over a-contamination for the joint of the response carriers. Two classes of estimates are treated: (1) Msestimates with bounded function p 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 oblta1rled as 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 "Sestimates" of regression [Robust and Nonlinear Time Series (1984): 256-272]. It is shown than an S-estimate based on a jump-function type p solves the min-max bias problem for the class of M-estimates with very general scale.