Combining locally and globally robust estimates for regression

Abstract

A new class of estimates for the linear model is introduced. These estimates, that we call C-estimates, are defined as a convex combination of a high breakdown point estimate, T 1 , and any other estimate, T 2 . We prove that C-estimates retain the global robustness properties of T 1 and inherit the local robustness behavior and the asymptotic distribution of T 2 . In particular, a C-estimate will have an asymptotic normal distribution and bounded contamination sensitivity if T 2 does. We also present an estimate with the maximum breakdown point and as efficient as the least squares estimate for normal errors. The maximum bias under outliers contamination of these estimates is computed for different fractions of contamination, and a Monte Carlo study is performed to asses the robustness and efficiency properties of the proposed estimates for finite sample size.