A Class of Locally and Globally Robust Regression Estimates

Abstract

Abstract We present a new class of regression estimates called generalized τ estimates. These estimates are defined by minimizing the τ scale of the weighted residuals, with weights that penalize high-leverage observations. Like the τ estimates, the generalized τ estimates utilize for their definition two loss functions, ρ1 and ρ2, which together with the weights can be chosen to achieve simultaneously high breakdown point, finite gross error sensitivity, and high efficiency. We recommend, however, choosing these functions so as to control the bias behavior of the estimate for a large range of possible contaminations and then boosting the efficiency by a simple least squares reweighting step. The generalized τ estimate with loss functions ρ1 and ρ2 is related to the Hill–Ryan GM estimate with a loss function ρ, which is a linear combination of ρ1 and ρr. In fact, both estimates have the same influence function and asymptotic distribution under the central model. We show that a certain generalized τ estimate has good maximum bias behavior and performs well in an extensive Monte Carlo simulation study and three numerical examples.