On the efficiency of Gini’s mean difference

Carina Gerstenberger () and Daniel Vogel

Statistical Methods & Applications, 2015, vol. 24, issue 4, 569-596

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 $$\epsilon $$ ϵ -fraction of a normal distribution with three times the standard deviation, the mean deviation is more efficient than the standard deviation—already for $$\epsilon > 1\,\%$$ ϵ > 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 efficient. We further compute the influence function of Gini’s mean difference, and carry out extensive finite-sample simulations. Copyright Springer-Verlag Berlin Heidelberg 2015

Keywords: Influence function; Mean deviation; Median absolute deviation; Normal mixture distribution; Residue theorem; Robustness; $$Q_n$$ Q n; Standard deviation; 62G35; 62G05; 62G20; C13 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (5)

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DOI: 10.1007/s10260-015-0315-x

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