ALTERNATIVES TO THE MEDIAN ABSOLUTE DEVIATION

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(i) {\x(i) - med(j)x(j)\}, because it has a simple explicit formula, needs little computation time, and is very robust a 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 S(n) = 1.1926 med(i) {med(j)\x(i) - x(j)} and the estimator Q(n) given by the.25 quantile of the distances {\x(i) - x(j)\i < j}. Note that S(n) and Q(n) do not need any location estimate. Both S(n) and Q(n) can be computed using O(n log n) time and O(n) storage. The Gaussian efficiency of S(n) is 58%, whereas Q(n) attains 82%. We study S(n) and Q(n) 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(n) has a lower gross-error sensitivity than the MAD.