The relationship between the absolute deviation from a quantile and Gini’s mean difference

We investigate the relationship between Gini’s mean difference (GMD), the mean absolute deviation, the least absolute deviation and the absolute deviation from a quantile. The latter can all be interpreted as equivalents either to the GMD of a transformed distribution or to a between-group GMD measure, according to the particular partition of the data. They all possess properties of the GMD but each omits the intra-group variability—and they give rise to different regression techniques. We argue that the analyst using one of these techniques should justify the omission of the intra-group variability from the analysis.

[1]  R. Koenker,et al.  Asymptotic Theory of Least Absolute Error Regression , 1978 .

[2]  Yusif Simaan.  Estimation risk in portfolio selection: the mean variance model versus the mean absolute deviation model , 1997 .

[3]  R. Koenker.  Quantile Regression: Name Index , 2005 .

[4]  A. Wald.  The Fitting of Straight Lines if Both Variables are Subject to Error , 1940 .

[5]  Z. Griliches,et al.  Is aggregation necessarily bad , 1960 .

[6]  Peter J. Lambert,et al.  The Gini Coefficient Reveals More , 2005 .

[7]  Thomas P. Hettmansperger,et al.  A robust analysis of the general linear model based on one step R-estimates , 1978 .

[8]  Jan Goebel,et al.  Using Analysis of Gini (ANOGI) for Detecting Whether Two Subsamples Represent the Same Universe , 2006 .

[9]  R. Carroll,et al.  Variance Function Estimation , 1987 .

[10]  Peter J. Huber,et al.  Robust Statistics , 2005, Wiley Series in Probability and Statistics.

[11]  Ingram Olkin,et al.  Gini Regression Analysis , 1992 .

[12]  H. Konno,et al.  Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market , 1991 .

[13]  Philippe C. Besse,et al.  A L 1-norm PCA and a Heuristic Approach , 1996 .

[14]  Shlomo Yitzhaki,et al.  Gini’s Mean difference: a superior measure of variability for non-normal distributions , 2003 .

[15]  S. Yitzhaki,et al.  On the proper bounds of the Gini correlation , 1999 .

[16]  Gini's Multiple Regressions: Two Approaches and Their Interaction , 2011 .

[17]  Edna Schechtman,et al.  A Measure of Association Based on Gini's Mean Difference , 1987 .

[18]  Vic Barnett,et al.  Outliers in Statistical Data , 1980 .

[19]  T. P. Hettmansperger,et al.  Statistical Inference Based on Ranks. , 1985 .

[20]  A. Pakes.  On the Asymptotic Bias of Wald-Type Estimators of a Straight Line When Both Variables Are Subject to Error , 1982 .

[21]  Liang Peng,et al.  Least absolute deviations estimation for ARCH and GARCH models , 2003 .

[22]  A. Shorrocks.  Ranking Income Distributions , 1983 .

[23]  S. Yitzhaki,et al.  The Gini Methodology , 2013 .

[24]  S. Yitzhaki.  Do we need a separate poverty measurement , 2002 .

[25]  I. Olkin,et al.  Concentration indices and concentration curves , 1991 .

[26]  R. Koenker,et al.  Robust Tests for Heteroscedasticity Based on Regression Quantiles , 1982 .

[27]  Jan Goebel,et al.  Using Analysis of Gini (Anogi) for Detecting Whether Two Sub-Samples Represent the Same Universe: The SOEP Experience , 2004, SSRN Electronic Journal.

[28]  S. Gorard.  REVISITING A 90-YEAR-OLD DEBATE: THE ADVANTAGES OF THE MEAN DEVIATION , 2005 .

[29]  L. Curtiss,et al.  ACCURATE DENSITY FUNCTIONAL THERMOCHEMISTRY FOR LARGER MOLECULES , 1997 .

[30]  Edna Schechtman,et al.  The Gini Methodology: A Primer on a Statistical Methodology , 2012 .