Reducing the mean squared error of quantile-based estimators by smoothing

Abstract

Many univariate robust estimators are based on quantiles. As already theoretically pointed out by Fernholz (in J. Stat. Plan. Inference 57(1), 29–38, 1997), smoothing the empirical distribution function with an appropriate kernel and bandwidth can reduce the variance and mean squared error (MSE) of some quantile-based estimators in small data sets. In this paper we apply this idea on several robust estimators of location, scale and skewness. We propose a robust bandwidth selection and bias reduction procedure. We show that the use of this smoothing method indeed leads to smaller MSEs, also at contaminated data sets. In particular, we obtain better performances for the medcouple which is a robust measure of skewness that can be used for outlier detection in skewed distributions.