Bahadur representations for the median absolute deviation and its modifications

Abstract

The median absolute deviation about the median (MAD) is an important univariate spread measure having wide appeal due to its highly robust sample version. A powerful tool in treating the asymptotics of a statistic is a linearization, i.e., a Bahadur representation. Here we establish both strong and weak Bahadur representations for the sample MAD. The strong version is the first such result in the literature, while the weak version improves upon previous treatments by reducing regularity conditions. Our results also apply to a modified version of sample MAD [Tyler, D.E., 1994. Finite sample breakdown points of projection based multivariate location and scatter statistics. Annals of Statistics 22, 1024–1044; Gather, U., Hilker, T., 1997. A note on Tyler’s modification of the MAD for the Stahel–Donoho estimator. Annals of Statistics 25, 2024–2026] introduced to obtain improved robustness for statistical procedures using sample median and MAD combinations over the univariate projections of multivariate data. The strong version yields the law of iterated logarithm for the sample MAD and supports the study of almost sure properties of randomly trimmed means based on the MAD and development of robust sequential nonparametric confidence intervals for the MAD. The weak version is needed to simplify derivations of the asymptotic joint distributions of vectors of dependent sample median and sample MAD combinations, which arise in constructing nonparametric multivariate outlyingness functions via projection pursuit.