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Assessing the difference between integrated quantiles and integrated cumulative distribution functions

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  • Yunran Wei
  • Ricardas Zitikis

Abstract

This paper offers a mathematical invention that shows how to convert integrated quantiles, which often appear in risk measures, into integrated cumulative distribution functions, which are technically more tractable from various perspectives. The invention helps to avoid a number of technical assumptions that have been traditionally imposed when working with quantities containing quantiles. In particular it helps to completely avoid the requirement of the existence of a probability density function. The developed results explain and illustrate the invention, whose byproducts include the assessment of model uncertainty and misspecification, and the derivation of statistical inference results.

Suggested Citation

  • Yunran Wei & Ricardas Zitikis, 2022. "Assessing the difference between integrated quantiles and integrated cumulative distribution functions," Papers 2210.16880, arXiv.org, revised Apr 2023.
    • Handle: RePEc:arx:papers:2210.16880
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    File URL: http://arxiv.org/pdf/2210.16880
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    References listed on IDEAS

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    1. Peng, Liang & Qi, Yongcheng & Wang, Ruodu & Yang, Jingping, 2012. "Jackknife empirical likelihood method for some risk measures and related quantities," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 142-150.
    2. Abdelhakim Necir & Abdelaziz Rassoul & Ričardas Zitikis, 2010. "Estimating the Conditional Tail Expectation in the Case of Heavy-Tailed Losses," Journal of Probability and Statistics, Hindawi, vol. 2010, pages 1-17, April.
    3. Goegebeur, Yuri & Guillou, Armelle & Pedersen, Tine & Qin, Jing, 2022. "Extreme-value based estimation of the conditional tail moment with application to reinsurance rating," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 102-122.
    4. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Post-Print hal-00413729, HAL.
    5. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496, December.
    6. 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.
    7. Song Xi Chen, 2008. "Nonparametric Estimation of Expected Shortfall," Journal of Financial Econometrics, Oxford University Press, vol. 6(1), pages 87-107, Winter.
    8. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Quantitative Finance, Taylor & Francis Journals, vol. 10(6), pages 593-606.
    9. Wang, Qiuqi & Wang, Ruodu & Wei, Yunran, 2020. "Distortion Riskmetrics On General Spaces," ASTIN Bulletin, Cambridge University Press, vol. 50(3), pages 827-851, September.
    10. Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional value-at-risk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1443-1471, July.
    11. N. V. Gribkova & J. Su & R. Zitikis, 2022. "Empirical tail conditional allocation and its consistency under minimal assumptions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(4), pages 713-735, August.
    12. Bellini, Fabio & Fadina, Tolulope & Wang, Ruodu & Wei, Yunran, 2022. "Parametric measures of variability induced by risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 270-284.
    13. Georg Ch Pflug & Werner Römisch, 2007. "Modeling, Measuring and Managing Risk," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 6478.
    14. Fabio Bellini & Tolulope Fadina & Ruodu Wang & Yunran Wei, 2020. "Parametric measures of variability induced by risk measures," Papers 2012.05219, arXiv.org, revised Apr 2022.
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    Cited by:

    1. N. V. Gribkova & J. Su & R. Zitikis, 2024. "Assessing the coverage probabilities of fixed-margin confidence intervals for the tail conditional allocation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 76(5), pages 821-850, October.

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