Alternatives to the Median Absolute Deviation

Abstract In robust estimation one frequently needs an initial or auxiliary estimate of scale. For this one usually takes the median absolute deviation MAD n = 1. 4826 med, |xi − med j x j |, because it has a simple explicit formula, needs little computation time, and is very robust as witnessed by its bounded influence function and its 50% breakdown point. But there is still room for improvement in two areas: the fact that MAD n is aimed at symmetric distributions and its low (37%) Gaussian efficiency. In this article we set out to construct explicit and 50% breakdown scale estimators that are more efficient. We consider the estimator Sn = 1. 1926 med, med j | xi − xj | and the estimator Qn given by the. 25 quantile of the distances |xi − x j |; i < j. Note that Sn and Qn do not need any location estimate. Both Sn and Qn can be computed using O (n log n) time and O (n) storage. The Gaussian efficiency of Sn is 58%, whereas Qn attains 82%. We study Sn and Qn by means of their influence functions, their bias curves (for implosion as well as explosion), and their finite-sample performance. Their behavior is also compared at non-Gaussian models, including the negative exponential model where Sn has a lower gross-error sensitivity than the MAD.