Exponential probability inequality and convergence results for the median absolute deviation and its modifications

The median absolute deviation from the median (MAD) is an important robust univariate spread measure. It also plays important roles with multivariate data through statistics based on the univariate projections of the data, in which case a modified sample MAD introduced by Tyler [Tyler, D. E, 1994. Finite sample breakdown points of projection based multivariate location and scatter statistics. Annals of Statistics 22, 1024-1044] and Gather and Hilker [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] is used to gain increased robustness. Here we establish for the modified sample MAD the same almost sure convergence to the population MAD as was shown by Hall and Welsh [Hall, P., Welsh, A. H., 1985. Limit theorems for the median deviation. Annals of Institute of Statistical Mathematics 37, 27-36] and Welsh [Welsh, A. H., 1986. Bahadur representations for robust scale estimators based on regression residuals. Annals of Statistics 14, 1246-1251] for the usual sample MAD, and at the same time we eliminate the regularity assumptions imposed in the previous results. Our method is to establish for the sample MAD and modified versions an exponential probability inequality which yields the desired almost sure convergence and also carries independent interest. Further, the asymptotic joint normality of the sample median and the sample MAD established by Falk [Falk, M., 1997a. On MAD and COMEDIANS. Annals of Institute of Statistical Mathematics 49, 615-644; Falk, M., 1997b. Asymptotic independence of median and MAD. Statistics and Probability Letters 34, 341-345] is extended to the modified sample MAD. Besides eliminating some regularity conditions, these results provide theoretical validation for use of the more general form of sample MAD.