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On the Lp-metric between a probability distribution and its distortion

Author

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  • López-Díaz, Miguel
  • Sordo, Miguel A.
  • Suárez-Llorens, Alfonso

Abstract

In actuarial theory, the Lp-metric is used to evaluate how well a probability distribution approximates another one. In the context of the distorted expectation hypothesis, the actuary replaces the original probability distribution by a distorted probability, so it makes sense to interpret the Lp-metric between them as a characteristic of the underlying random variable. We show in this paper that this is a characteristic of the variability of the random variable, study its properties and give some applications.

Suggested Citation

  • López-Díaz, Miguel & Sordo, Miguel A. & Suárez-Llorens, Alfonso, 2012. "On the Lp-metric between a probability distribution and its distortion," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 257-264.
    • Handle: RePEc:eee:insuma:v:51:y:2012:i:2:p:257-264
    DOI: 10.1016/j.insmatheco.2012.04.004
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    References listed on IDEAS

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    1. Miguel Sordo & Héctor Ramos, 2007. "Characterization of stochastic orders by L-functionals," Statistical Papers, Springer, vol. 48(2), pages 249-263, April.
    2. Belzunce, Félix & Suárez-Llorens, Alfonso & Sordo, Miguel A., 2012. "Comparison of increasing directionally convex transformations of random vectors with a common copula," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 385-390.
    3. Denneberg, Dieter, 1990. "Premium Calculation: Why Standard Deviation Should be Replaced by Absolute Deviation1," ASTIN Bulletin, Cambridge University Press, vol. 20(2), pages 181-190, November.
    4. Wang, Shaun, 1996. "Premium Calculation by Transforming the Layer Premium Density," ASTIN Bulletin, Cambridge University Press, vol. 26(1), pages 71-92, May.
    5. Shaun Wang, 1998. "An Actuarial Index of the Right-Tail Risk," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(2), pages 88-101.
    6. Ramos, Héctor M. & Sordo, Miguel A., 2003. "Dispersion measures and dispersive orderings," Statistics & Probability Letters, Elsevier, vol. 61(2), pages 123-131, January.
    7. 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.
    8. Darolles, Serge & Gourieroux, Christian & Jasiak, Joann, 2009. "L-performance with an application to hedge funds," Journal of Empirical Finance, Elsevier, vol. 16(4), pages 671-685, September.
    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. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    11. Shaun, Wang, 1995. "Insurance pricing and increased limits ratemaking by proportional hazards transforms," Insurance: Mathematics and Economics, Elsevier, vol. 17(1), pages 43-54, August.
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    Citations

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    Cited by:

    1. Gómez-Déniz, Emilio & Sordo, Miguel A. & Calderín-Ojeda, Enrique, 2014. "The Log–Lindley distribution as an alternative to the beta regression model with applications in insurance," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 49-57.
    2. repec:bpj:demode:v:6:y:2018:i:1:p:156-177:n:10 is not listed on IDEAS
    3. 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.
    4. 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.
    5. Alois Pichler, 2017. "A quantitative comparison of risk measures," Annals of Operations Research, Springer, vol. 254(1), pages 251-275, July.
    6. Sordo, M.A. & Bello, A.J. & Suárez-Llorens, A., 2018. "Stochastic orders and co-risk measures under positive dependence," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 105-113.
    7. Hu, Taizhong & Chen, Ouxiang, 2020. "On a family of coherent measures of variability," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 173-182.
    8. Yang, Jianping & Hu, Taizhong, 2016. "New developments on the Lp-metric between a probability distribution and its distortion," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 236-243.
    9. Sánchez-Sánchez, M. & Sordo, M.A. & Suárez-Llorens, A. & Gómez-Déniz, E., 2019. "Deriving Robust Bayesian Premiums Under Bands Of Prior Distributions With Applications," ASTIN Bulletin, Cambridge University Press, vol. 49(1), pages 147-168, January.

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