An overview of asymptotic properties of Lp regression under general classes of error distributions

Abstract

We survey the asymptotic properties of regression Lp estimators under general classes of error distributions. It is found that the asymptotic distributions of Lp estimators depend crucially on p and the shape of the error distribution near the origin. A number of important features arise as a result, among which are (a) use of a small p may yield accelerated convergence rates for Lp estimators under certain classes of error distributions; (b) for p < 1, Lp regression should, under some circumstances, be undertaken by locally maximizing, rather than minimizing, the sum of the pth powers of the absolute deviations; and (c) consistent estimation of the sampling distributions of the Lp estimators can be achieved by the m out of n bootstrap in general. Numerical examples are provided to illustrate our theoretical findings, and a computational algorithm is suggested for local maximization as may sometimes be required by the Lp procedure. © 2005 American Statistical Association.