# On the efficiency of Gini’s mean difference

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## Date

2015

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## Abstract

The asymptotic relative efficiency of the mean deviation with respect to the standard
deviation is 88% at the normal distribution. In his seminal 1960 paper A survey of sampling from
contaminated distributions, J. W. Tukey points out that, if the normal distribution is contaminated
by a small c-fraction of a normal distribution with three times the standard deviation, the mean
deviation is more efficient than the standard deviation - already for c < 1%. In the present
article, we examine the efficiency of Gini's mean difference (the mean of all pairwise distances).
Our results may be summarized by saying Gini's mean difference combines the advantages of the
mean deviation and the standard deviation. In particular, an analytic expression for the finite-
sample variance of Gini's mean difference at the normal mixture model is derived by means of
the residue theorem, which is then used to determine the contamination fraction in Tukey's 1:3
normal mixture distribution that renders Gini's mean difference and the standard deviation equally
effcient. We further compute the in influence function of Gini's mean difference, and carry out
extensive finite-sample simulations.

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## Keywords

influence function, standard deviation, Qn, robustness, residue theorem, normal mixture distribution, median absolute deviation, mean deviation