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Shape measures based on the convex transform order

Author

Listed:
  • A. Arriaza

    (Universidad de Cádiz)

  • A. Crescenzo

    (Università di Salerno)

  • M. A. Sordo

    (Universidad de Cádiz)

  • A. Suárez-Llorens

    (Universidad de Cádiz)

Abstract

Three functional measures of the shape of univariate distributions are proposed which are consistent with respect to the convex transform order. The first two are weighted tail indices that characterize location-scale families of distributions, whilst the third is a skewness measure. Properties of the new measures are established for various classes of symmetric and asymmetric distributions, and the generalized Pareto distribution characterized in terms of them. Kernel density based estimation of the measures is also considered, and the use of the estimated functionals is illustrated in the analysis of two real data sets.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metrik:v:82:y:2019:i:1:d:10.1007_s00184-018-0667-y
    DOI: 10.1007/s00184-018-0667-y
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    References listed on IDEAS

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    1. Jones, M. C. & Rosco, J. F. & Pewsey, Arthur, 2011. "Skewness-Invariant Measures of Kurtosis," The American Statistician, American Statistical Association, vol. 65(2), pages 89-95.
    2. Peter H. Westfall, 2014. "Kurtosis as Peakedness, 1905-2014. R.I.P," The American Statistician, Taylor & Francis Journals, vol. 68(3), pages 191-195, April.
    3. Boshnakov, Georgi N., 2007. "Some measures for asymmetry of distributions," Statistics & Probability Letters, Elsevier, vol. 77(11), pages 1111-1116, June.
    4. 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.
    5. Félix Belzunce & Julio Mulero & José María Ruíz & Alfonso Suárez-Llorens, 2015. "On relative skewness for multivariate distributions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(4), pages 813-834, December.
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    Cited by:

    1. Abdolsaeed Toomaj & Antonio Di Crescenzo, 2020. "Connections between Weighted Generalized Cumulative Residual Entropy and Variance," Mathematics, MDPI, vol. 8(7), pages 1-27, July.

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