Bootrapping robust estimates of regression

TLDR

The paper introduces a computer‑intensive bootstrap method to estimate the distribution of robust regression estimates. The method bootstraps a reweighted representation of robust regression estimates, incorporating an auxiliary scale estimate, using weights that decrease with residual magnitude, and requires only solving a linear system for each bootstrap sample. The bootstrap method is resistant to outliers, yields higher breakdown points for quantile estimates than conventional bootstrap, and its performance was demonstrated on two datasets and a Monte Carlo study of confidence intervals.

Abstract

We introduce a new computer-intensive method to estimate the distribution of robust regression estimates. The basic idea behind our method is to bootstrap a reweighted representation of the estimates. To obtain a bootstrap method that is asymptotically correct, we include the auxiliary scale estimate in our reweighted representation of the estimates. Our method is computationally simple because for each bootstrap sample we only have to solve a linear system of equations. The weights we use are decreasing functions of the absolute value of the residuals and hence outlying observations receive small weights. This results in a bootstrap method that is resistant to the presence of outliers in the data. The breakdown points of the quantile estimates derived with this method are higher than those obtained with the bootstrap. We illustrate our method on two datasets and we report the results of a Monte Carlo experiment on confidence intervals for the parameters of the linear model.