Robust and efficient estimation of the residual scale in linear regression

Robustness and efficiency of the residual scale estimators in the regression model is important for robust inference. We introduce the class of robust generalized M-scale estimators for the regression model, derive their influence function and gross-error sensitivity, and study their maxbias behavior. In particular, we find overall minimax bias estimates for the general class and also for well-known subclasses. We pose and solve a Hampel’s-like optimality problem: we find generalized M-scale estimators with maximal efficiency subject to a lower bound on the global and local robustness of the estimators.