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On the efficiency of Gini’s mean difference

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  • Carina Gerstenberger
  • Daniel Vogel

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

Suggested Citation

  • Carina Gerstenberger & Daniel Vogel, 2015. "On the efficiency of Gini’s mean difference," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(4), pages 569-596, November.
    • Handle: RePEc:spr:stmapp:v:24:y:2015:i:4:p:569-596
    DOI: 10.1007/s10260-015-0315-x
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    References listed on IDEAS

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    1. Shlomo Yitzhaki, 2003. "Gini’s Mean difference: a superior measure of variability for non-normal distributions," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(2), pages 285-316.
    2. Gutti Babu & C. Rao, 1992. "Expansions for statistics involving the mean absolute deviations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 44(2), pages 387-403, June.
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    Cited by:

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    2. Maria-Teresa Bosch-Badia & Joan Montllor-Serrats & Maria-Antonia Tarrazon-Rodon, 2017. "Analysing assets’ performance inside a portfolio: From crossed beta to the net risk premium ratio," Cogent Economics & Finance, Taylor & Francis Journals, vol. 5(1), pages 1270251-127, January.
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    4. Pierpaolo Angelini & Fabrizio Maturo, 2023. "Tensors Associated with Mean Quadratic Differences Explaining the Riskiness of Portfolios of Financial Assets," JRFM, MDPI, vol. 16(8), pages 1-25, August.
    5. Fabrizio Maturo & Pierpaolo Angelini, 2023. "Aggregate Bound Choices about Random and Nonrandom Goods Studied via a Nonlinear Analysis," Mathematics, MDPI, vol. 11(11), pages 1-30, May.

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    More about this item

    Keywords

    Influence function; Mean deviation; Median absolute deviation; Normal mixture distribution; Residue theorem; Robustness; $$Q_n$$ Q n ; Standard deviation; 62G35; 62G05; 62G20; C13;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General

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