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, {|x_i − 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 S_n = 1.1926 med, {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.

Key Words:

• Bias curve
• Breakdown point
• Influence function
• Robustness
• Scale estimation