A measure of asymmetry

Statistical Papers volume 53, pages971–985(2012)Cite this article

Abstract

It is a general practice to make assertions about the symmetry or asymmetry of a probability density function based on the coefficients of skewness. Since most of the coefficients of skewness are designed to be zero for a symmetric density, they overall do provide an indication of symmetry. However, skewness is primarily influenced by the tail behavior of a density function, and the skewness coefficients are designed to capture this behavior. Thus they do not calibrate asymmetry in the density curves. We provide a necessary condition for a probability density function to be symmetric and use that to measure asymmetry in a continuous density curve on the scale of −1 to 1. We show through examples that the proposed measure does an admirable job of capturing the visual impression of asymmetry of a continuous density function.

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References

  1. Arias-Nicolás JP, Martín J, Suárez-Llorens A (2009) L p loss functions: a robust bayesian approach. Stat Papers 50: 501–509

    MATH  Article  Google Scholar 

  2. Boos DD (1982) A test for asymmetry associated with the Hodges-Lehmann estimator. J Am Stat Assoc 77: 647–649

    MathSciNet  MATH  Article  Google Scholar 

  3. Boshnakov GN (2007) Some measures for asymmetry of distributions. Stat Prob Lett 77: 1111–1116

    MathSciNet  MATH  Article  Google Scholar 

  4. Butler CC (1969) A test for symmetry using the sample distribution function. Ann Math Stat 40: 2209–2210

    MATH  Article  Google Scholar 

  5. Critchley F, Jones MC (2008) Asymmetry and gradient asymmetry functions: density-based skewness and kurtosis. Scand J Stat 35: 415–437

    MathSciNet  MATH  Article  Google Scholar 

  6. Doksum KA (1975) Measures of location and asymmetry. Scand J Stat 1: 11–22

    MathSciNet  Google Scholar 

  7. Ekström M, Jammalamadaka SR (2007) An asymptotically distribution-free test of symmetry. J Stat Plan Inference 137: 799–810

    MATH  Article  Google Scholar 

  8. Giné E, Mason D (2008) Uniform in Bandwidth Estimation of Integral Functionals of the Density Function. Scand J Stat 35: 739–761

    MATH  Article  Google Scholar 

  9. Jones MC (1993) Simple boundary correction for kernel density estimation. Stat Comput 3: 135–146

    Article  Google Scholar 

  10. Li X, Morris JM (1991) On measuring asymmetry and the reliability of the skewness measure. Stat Prob Lett 12: 267–271

    MathSciNet  Article  Google Scholar 

  11. MacGillivray HL (1986) Skewness and asymmetry: measures and orderings. Annals Stat 14: 994–1011

    MathSciNet  MATH  Article  Google Scholar 

  12. Rothman ED, Woodroofe M (1972) A Cramé-von Mises type statistic for testing symmetry. Ann Math Stat 43: 2035–2038

    MathSciNet  MATH  Article  Google Scholar 

  13. Siburg KF, Stoimenov PA (2008) Symmetry of functions and exchangeability of random variables. Stat Papers. doi:10.1007/s00362-008-0195-3

  14. Ye P, Bhattacharya B (2009) Tests of symmetry with one-sided alternatives in three-way contingency tables. Stat Papers. doi:10.1007/s00362-009-0198-8

  15. van Zwet WR (1964) Convex transformations of random variables. Math Centrum, Amsterdam

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Affiliations

  1. School of Mathematics and Statistics, The University of Birmingham, Birmingham, B15 2TT, UK

    P. N. Patil

  2. Department of Statistics, Miami University, Oxford, OH, 45056, USA

    P. P. Patil

  3. Accenture, 1 Arkadias Street, 14564, Athens, Greece

    D. Bagkavos

Authors
  1. P. N. Patil
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  2. P. P. Patil
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  3. D. Bagkavos
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Correspondence to D. Bagkavos.

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Patil, P.N., Patil, P.P. & Bagkavos, D. A measure of asymmetry. Stat Papers 53, 971–985 (2012). https://doi.org/10.1007/s00362-011-0401-6

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Keywords

  • Asymmetry measure
  • Correlation
  • Nonparametric
  • Skewness