Article,
Alternatives to the Median Absolute Deviation
Journal of the American Statistical Association, (1993)
DOI: 10.2307/2291267
Abstract
In robust estimation one frequently needs an initial or auxiliary estimate of scale. For this one usually takes the median absolute deviation <tex-math>\$MAD\_n = 1.4826 med\_i\| x\_i - med\_jx\_j|\\$</tex-math>, 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<sub>n</sub> 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 <tex-math>\$S\_n = 1.1926 med\_i\med\_j|x\_i - x\_j|\\$</tex-math> and the estimator Q<sub>n</sub> given by the .25 quantile of the distances <latex>\$\|x\_i - x\_j|; i < j\\$</latex>. Note that S<sub>n</sub> and Q<sub>n</sub> do not need any location estimate. Both S<sub>n</sub> and Q<sub>n</sub> can be computed using O(n log n) time and O(n) storage. The Gaussian efficiency of S<sub>n</sub> is 58\%, whereas Q<sub>n</sub> attains 82\%. We study S<sub>n</sub> and Q<sub>n</sub> 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 S<sub>n</sub> has a lower gross-error sensitivity than the MAD.
