A Comparative Study of Robust and Stable Estimates of Multivariate Location

Abstract

This work is concerned with the comparative analysis of a variety of robust estimates of location under the generalized Gaussian and the Student t- and the Tukey gross-error distributions in the univariate and multivariate cases. The chosen set of estimates comprises the sample mean, sample median, classical robust Maronna, Huber, and Hampel M-estimates, Meshalkin–Shurygin stable M-estimates with redescending score functions, and a low-complexity two-step estimate with the preliminary rejection of outliers by the Tukey boxplot rule followed by the use of the sample mean to the cleaned data — almost all of them are examined in the univariate and multivariate versions. The estimate performance is evaluated by efficiency, bias, and mean squared error. For univariate distributions with light and heavy tails, the best results are exhibited by the Huber, Hampel, and Meshalkin–Shurygin and two-step estimates of location. In the multivariate case, the Huber and two-step estimates perform best.