M-estimation in linear models under nonstandard conditions

The limiting distribution of M-estimators of the regression parameter in linear models is derived under nonstandard conditions, allowing, e.g., for discontinuities in density functions. Unlike usual regularity assumptions, our conditions are satisfied, for instance, in the case of regression quantiles, hence also in the context of L-1 estimation; our results thus extend those of Knight (Ann. Statist. 26 (1998) 755). The resulting asymptotic distributions, in general, are not Gaussian. Therefore, the limiting bootstrap distributions of these estimators are also investigated. It is shown that bootstrap approximations are correct to the first order only when limiting distributions are Gaussian, or along specific sequences m, of bootstrap sample sizes. Numerical examples are given to illustrate these asymptotic results. (C) 2003 Elsevier B.V. All rights reserved.