1993/12/01 by Peter J. Rousseeuw, Christophe Croux · 1,477 citations
Decision Sciences · Mathematics · #Absolute deviation #Advanced Statistical Methods and Models #Advanced Statistical Process Monitoring #Applied mathematics #Bounded function #Combinatorics #Estimator #Exponential function #Function (biology) #Gaussian #Least absolute deviations #Mathematical analysis #Mathematics #Physics #Quantile #Statistical Distribution Estimation and Applications #Statistics
paper · doi:10.2307/2291267
published in Journal of the American Statistical Association 88(424), 1273
openalex publication_date 1993/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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.