Robust Minimax Variance Estimation of Location under Bounded Distribution Interquantile Ranges

Abstract

The performance of the minimax variance (in the Huber sense) robust M-estimates of a location parameter designed in the class of symmetric distributions with bounded interquantile ranges is studied both on large and small samples. Although the general structure of these minimax M-estimates has been described by Huber (1981) long ago, they were not studied as main attention was focused on the estimates designed for the various neighborhoods of a Gaussian. In this work, the proposed estimates are compared to the sample mean, sample median, and the classical robust Huber and Hampel M-estimates under the Gaussian, contaminated Gaussian, Laplace, and Cauchy distributions. The performance of an estimate is measured by its efficiency, bias, and mean squared error. The following conclusions are made: (i) under the Gaussian, Laplace, Cauchy, and moderately contaminated Gaussian distributions, the proposed minimax M-estimates outperform the robust Huber and Hampel M-estimates with respect to asymptotic efficiency; (ii) under heavily contaminated Gaussian distributions, the Huber and Hampel M-estimates are slightly better; (iii) on small samples, these classical robust estimates also slightly outperform the proposed minimax estimates in mean squared error.