Approximation to the distributions of M-estimates in linear models by randomly weighted bootstrap

We consider the M-estimation of regression parameters in the linear model by minimizing the sum of convex functions of residuals. In earlier papers (see for instance Bai, Rao and Wu (1992) and Yohai and Maronna (1979)); the asymptotic normaility of the M-estimator was established. In this paper we discuss the method of Bayesian bootstrap to derive the approximate distribution of the M-estimator. Bayesian bootstrap or the random weighting method was developed by Rubin (1981), Lo (1987), Weng(1989), Zheng(1987) and Tu and Zheng (1987) with reference to some statistics such as the sample mean. We extend these results to the general regression problem