Statistica Sinica 18(2008), 1619-1640

RATEWISE EFFICIENT ESTIMATION OF REGRESSION

COEFFICIENTS BASED ON $\mbox{\boldmath $L_p$}$ PROCEDURES

P. Y. Lai and Stephen M. S. Lee

The University of Hong Kong

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 $L_p$ estimators based on judicious choices of $p$. We develop a procedure for choosing $p$ adaptively to yield $L_p$ estimators that converge at approximately optimal rates. The procedure consists of a special algorithm to automatically select the correct mode of $L_p$ estimation and the $m$ out of $n$ bootstrap to consistently estimate the log mean squared error of the $L_p$ estimator. Our proposed adaptive $L_p$ estimator is compared with other adaptive and non-adaptive $L_p$ estimators in a simulation study, that confirms superiority of our procedure.

Key words and phrases: Adaptive, $L_p$ estimator, $m$ out of $n$ bootstrap, ratewise efficient, regression.