Abstract: The median absolute deviation about the median (MAD) is an important robust univariate spread measure. In applications with multivariate data using projection pursuit and (Median, MAD) combinations for the univariate projections, a modified version of sample MAD (Tyler, 1994, Gather and Hilker, 1997) yields increased robustness. Here we establish for the modified MAD the same almost sure convergence to population MAD shown by Hall and Welsh (1985) for the usual sample MAD, and at the same time we eliminate the previous regularity conditions. Our method is to establish an exponential probability inequality for the sample MAD (both usual and modified versions). Further, joint asymptotic normality of the sample median and modified sample MAD is shown. Also, eliminating a symmetry assumption of Hall and Welsh (1985), we establish almost sure and in-probability type Bahadur representations for the sample MAD and modified versions. Several applications will be noted.

(Joint Work with Professor Robert Serfling).