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ON MAD AND COMEDIANS

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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-*alpha*-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.

**Source**
( TeX ,
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