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Asymmetry and Gradient Asymmetry Functions: Density‐Based Skewness and Kurtosis

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  • FRANK CRITCHLEY
  • M. C. JONES

Abstract

. Functional measures of skewness and kurtosis, called asymmetry and gradient asymmetry functions, are described for continuous univariate unimodal distributions. They are defined and interpreted directly in terms of the density function and its derivative. Asymmetry is defined by comparing distances from points of equal density to the mode. Gradient asymmetry is defined, in novel fashion, as asymmetry of an appropriate function of the density derivative. Properties and illustrations of asymmetry and gradient asymmetry functions are presented. Estimation of them is considered and illustrated with an example. Scalar summary skewness and kurtosis measures associated with asymmetry and gradient asymmetry functions are discussed.

Suggested Citation

  • Frank Critchley & M. C. Jones, 2008. "Asymmetry and Gradient Asymmetry Functions: Density‐Based Skewness and Kurtosis," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 415-437, September.
    • Handle: RePEc:bla:scjsta:v:35:y:2008:i:3:p:415-437
    DOI: 10.1111/j.1467-9469.2008.00599.x
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    References listed on IDEAS

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    1. Boshnakov, Georgi N., 2007. "Some measures for asymmetry of distributions," Statistics & Probability Letters, Elsevier, vol. 77(11), pages 1111-1116, June.
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    4. Robert Serfling, 2002. "Quantile functions for multivariate analysis: approaches and applications," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 56(2), pages 214-232, May.
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    1. Giri Parameswaran & Hunter Rendleman, 2022. "Redistribution under general decision rules," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 24(1), pages 159-196, February.
    2. Robert Staudte, 2014. "Inference for quantile measures of skewness," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(4), pages 751-768, December.
    3. Chen, Yan & Liu, Yakun, 2024. "Idiosyncratic asymmetry in stock returns: An entropy measure," Finance Research Letters, Elsevier, vol. 64(C).
    4. Andreas Eberl & Bernhard Klar, 2022. "Expectile‐based measures of skewness," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 373-399, March.
    5. Andreas Eberl & Bernhard Klar, 2021. "A note on a measure of asymmetry," Statistical Papers, Springer, vol. 62(3), pages 1483-1497, June.
    6. Xu, Zhongxiang & Chevapatrakul, Thanaset & Li, Xiafei, 2019. "Return asymmetry and the cross section of stock returns," Journal of International Money and Finance, Elsevier, vol. 97(C), pages 93-110.
    7. Christophe Ley, 2014. "Flexible Modelling in Statistics: Past, present and Future," Working Papers ECARES ECARES 2014-42, ULB -- Universite Libre de Bruxelles.
    8. M. C. Jones, 2015. "On Families of Distributions with Shape Parameters," International Statistical Review, International Statistical Institute, vol. 83(2), pages 175-192, August.
    9. Rubio, Francisco Javier & Steel, Mark F. J., 2014. "Bayesian modelling of skewness and kurtosis with two-piece scale and shape transformations," MPRA Paper 57102, University Library of Munich, Germany.
    10. P. Patil & P. Patil & D. Bagkavos, 2012. "A measure of asymmetry," Statistical Papers, Springer, vol. 53(4), pages 971-985, November.
    11. Gurjeet Dhesi & Bilal Shakeel & Marcel Ausloos, 2021. "Modelling and forecasting the kurtosis and returns distributions of financial markets: irrational fractional Brownian motion model approach," Annals of Operations Research, Springer, vol. 299(1), pages 1397-1410, April.
    12. A. Arriaza & A. Crescenzo & M. A. Sordo & A. Suárez-Llorens, 2019. "Shape measures based on the convex transform order," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(1), pages 99-124, January.
    13. J. Rosco & M. Jones & Arthur Pewsey, 2011. "Skew t distributions via the sinh-arcsinh transformation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(3), pages 630-652, November.
    14. Baillien, Jonas & Gijbels, Irène & Verhasselt, Anneleen, 2023. "A new distance based measure of asymmetry," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    15. Irène Gijbels & Rezaul Karim & Anneleen Verhasselt, 2020. "Response to the Letter to the Editor on ‘On Quantile‐based Asymmetric Family of Distributions: Properties and Inference’," International Statistical Review, International Statistical Institute, vol. 88(3), pages 797-801, December.

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