Robust estimates and tests for the one- and two-sample scale models

Abstract

The purpose of this paper is to investigate the scale problem and develop a class of robust techniques which are useful in estimation and testing. The classical functional which has been used as a measure of variability of a distribution is the standard deviation. Its associated estimator, the sample standard deviation, is shown to have highly undesirable robustness properties. Another measure of variability, the square root of the midvariance, is proposed and investigated. The midvariance is basically the asymptotic variance of an M -estimator of location and is shown to behave like a Winsorized variance. The asymptotic distribution of the sample estimator of the midvariance is found and a small-sample approximation of the distribution using the chi-squared distribution is developed. With the above machinery, one-sample and two-sample tests of variability are constructed.