Ratewise efficient estimation of regression coefficients based on Lp procedures

Abstract

We consider the problem of estimation of regression coefficients under general classes of error densities without assuming classical regularity conditions. Optimal orders of convergence rates of regression-equivariant estimators are established and shown to be attained in general by Lp estimators based on judicious choices of p. We develop a procedure for choosing p adaptively to yield Lp estimators that converge at approximately optimal rates. The procedure consists of a special algorithm to automatically select the correct mode of Lp estimation and the m out of n bootstrap to consistently estimate the log mean squared error of the Lp estimator. Our proposed adaptive Lp estimator is compared with other adaptive and non-adaptive Lp estimators in a simulation study, that confirms superiority of our procedure.