Models Where the Least Trimmed Squares and Least Median of Squares Estimators Are Maximum Likelihood

Abstract

The Least Trimmed Squares (LTS) and Least Median of Squares (LMS) estimators are popular robust regression estimators. The idea behind the estimators is to find, for a given h, a sub-sample of h 'good' observations among n observations and estimate the regression on that sub-sample. We find models, based on the normal or the uniform distribution respectively, in which these estimators are maximum likelihood. We provide an asymptotic theory for the location-scale case in those models. The LTS estimator is found to be h1/2 consistent and asymptotically standard normal. The LMS estimator is found to be h consistent and asymptotically Laplace.