ON MAD AND COMEDIANS

MICHAEL FALK

Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt, Ostenstrasse 26–28, D-85071 Eichstätt, Germany

(Received March 8, 1996; revised October 15, 1996)

Abstract. A popular robust measure of dispersion of a random variable (rv)

X is the median absolute deviation from the median med(|X − med(X)|),

MAD for short, which is based on the median med(X) of X. By choosing Y = X, the MAD turns out to be a special case of the comedian med((X − med(X))(Y − med(Y))), which is a robust measure of covariance between rvs X and Y. We investigate the comedian in detail, in particular in the normal case, and establish strong consistency and asymptotic normality of empirical counterparts. This leads to a robust competitor of the coefficient of correlation as an asymptotic level-α-statistic for testing independence of X and Y. An example shows the weird fact that knowledge of the population med(X) does not necessarily pay (in the sense of asymptotic relative efficiency) when estimating the MAD.

Key words and phrases: Median absolute deviation from the median, robust measure of correlation, comedian, breakdown point, covariance, correlation coefficient, bivariate normal vectors, strong consistency, asymptotic normality.