Extended Mean with Distribution-Free Variance

Abstract

Location estimation is one of the basic activities in statistical data analysis so considerable effort has been put into the development of procedures for the robust estimation of measures of location. Because the distribution-free variance of most of existing measures is difficult to obtain in closed form, these measures work under strong modelling assumptions. We propose a robust location measure in which the expectation of a lower order statistics is replaced by the expectation of a larger order statistics. The main attraction of this measure is that its distribution-free variance is obtained in closed form. Comparisons with some of the best location estimators, mean, Hodges-Lehmann estimator, Huber's M-estimator and median are given based on Monte Carlo simulations. Computationally, the new estimator has an explicit expression and requires no iteration.