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Stochastic Comparisons of Some Distances between Random Variables

Author

Listed:
  • Patricia Ortega-Jiménez

    (Departamento de Estadística e I. O., Facultad de Ciencias, Universidad de Cádiz, 11002 Cádiz, Spain)

  • Miguel A. Sordo

    (Departamento de Estadística e I. O., Facultad de Ciencias, Universidad de Cádiz, 11002 Cádiz, Spain)

  • Alfonso Suárez-Llorens

    (Departamento de Estadística e I. O., Facultad de Ciencias, Universidad de Cádiz, 11002 Cádiz, Spain)

Abstract

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

Suggested Citation

  • Patricia Ortega-Jiménez & Miguel A. Sordo & Alfonso Suárez-Llorens, 2021. "Stochastic Comparisons of Some Distances between Random Variables," Mathematics, MDPI, vol. 9(9), pages 1-14, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:9:p:981-:d:544609
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    References listed on IDEAS

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    1. Balakrishnan, Narayanaswamy & Belzunce, Félix & Sordo, Miguel A. & Suárez-Llorens, Alfonso, 2012. "Increasing directionally convex orderings of random vectors having the same copula, and their use in comparing ordered data," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 45-54.
    2. Wang, Shaun, 1996. "Premium Calculation by Transforming the Layer Premium Density," ASTIN Bulletin, Cambridge University Press, vol. 26(1), pages 71-92, May.
    3. Young, Virginia R., 1999. "Discussion of Christofides' Conjecture Regarding Wang's Premium Principle," ASTIN Bulletin, Cambridge University Press, vol. 29(2), pages 191-195, November.
    4. Furman, Edward & Wang, Ruodu & Zitikis, Ričardas, 2017. "Gini-type measures of risk and variability: Gini shortfall, capital allocations, and heavy-tailed risks," Journal of Banking & Finance, Elsevier, vol. 83(C), pages 70-84.
    5. Sordo, Miguel A. & Suárez-Llorens, Alfonso, 2011. "Stochastic comparisons of distorted variability measures," Insurance: Mathematics and Economics, Elsevier, vol. 49(1), pages 11-17, July.
    6. 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.
    7. Sordo, Miguel A. & Castaño-Martínez, Antonia & Pigueiras, Gema, 2016. "A family of premium principles based on mixtures of TVaRs," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 397-405.
    8. Bassan, Bruno & Denuit, Michel & Scarsini, Marco, 1999. "Variability orders and mean differences," Statistics & Probability Letters, Elsevier, vol. 45(2), pages 121-130, November.
    9. R. Rockafellar & Stan Uryasev & Michael Zabarankin, 2006. "Generalized deviations in risk analysis," Finance and Stochastics, Springer, vol. 10(1), pages 51-74, January.
    10. Jorge Navarro & Yolanda del Águila & Miguel A. Sordo & Alfonso Suárez‐Llorens, 2013. "Stochastic ordering properties for systems with dependent identically distributed components," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 29(3), pages 264-278, May.
    11. Christian Genest & Jean‐François Quessy & Bruno Rémillard, 2006. "Goodness‐of‐fit Procedures for Copula Models Based on the Probability Integral Transformation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 337-366, June.
    12. Belzunce, Félix & Ruiz, José M. & Suárez-Llorens, Alfonso, 2008. "On multivariate dispersion orderings based on the standard construction," Statistics & Probability Letters, Elsevier, vol. 78(3), pages 271-281, February.
    13. Ortega-Jiménez, P. & Sordo, M.A. & Suárez-Llorens, A., 2021. "Stochastic orders and multivariate measures of risk contagion," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 199-207.
    14. Chen Li & Xiaohu Li, 2018. "Preservation of increasing convex/concave order under the formation of parallel/series system of dependent components," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(4), pages 445-464, May.
    15. Block, Henry W. & Savits, Thomas H. & Shaked, Moshe, 1985. "A concept of negative dependence using stochastic ordering," Statistics & Probability Letters, Elsevier, vol. 3(2), pages 81-86, April.
    16. Kochar, Subhash C. & Carrière, K. C., 1997. "Connections among various variability orderings," Statistics & Probability Letters, Elsevier, vol. 35(4), pages 327-333, November.
    17. Sordo, Miguel A., 2009. "Comparing tail variabilities of risks by means of the excess wealth order," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 466-469, December.
    18. Cai, Jun & Wei, Wei, 2012. "On the invariant properties of notions of positive dependence and copulas under increasing transformations," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 43-49.
    19. Giovagnoli, Alessandra & Wynn, H. P., 1995. "Multivariate dispersion orderings," Statistics & Probability Letters, Elsevier, vol. 22(4), pages 325-332, March.
    20. Yang, Jianping & Zhuang, Weiwei & Hu, Taizhong, 2014. "Lp-metric under the location-independent risk ordering of random variables," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 321-324.
    21. Sordo, Miguel A., 2008. "Characterizations of classes of risk measures by dispersive orders," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 1028-1034, June.
    22. Sordo, Miguel A. & Suárez-Llorens, Alfonso & Bello, Alfonso J., 2015. "Comparison of conditional distributions in portfolios of dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 62-69.
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