APPROXIMATING THE DISTRIBUTION OF M-ESTIMATORS IN LINEAR MODELS BY RANDOMLY WEIGHTED BOOTSTRAP

Abstract

The asymptotic distribution of the M-estimators are generally related to quan- tities of the error distribution that can not be conveniently estimated.The randomly weighted bootstrap method provides a way of assessing the distribution of the M-estimators without estimating the nuisance quantities of the error distributions.In this paper,the distribution of M-estimators is approximated by the randomly weighted bootstrap method in linear models when the covariates are random.It is shown that the randomly weighted bootstrapping estima- tion of the distribution of the M-estimator is uniformly consistent.Also,the variance estimates is investigated by Monte Carlo simulations for different choices of the convex function,sample size and random weights.Poisson weighting is recommended for reducing the computational burden in the randomly weighted bootstrapping M-estimators.