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Distortion risk measure under parametric ambiguity

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  • Shao, Hui
  • Zhang, Zhe George

Abstract

This study develops closed-form solutions for distortion risk measures (DRM) in extreme cases by utilizing the first two moments and the symmetry of underlying distributions. The resultant extreme-case distributions, encompassing the worst- and best-case distributions, are identified by the envelopes of the distortion functions. The findings of this study extend previous research on worst-case risk measures such as worst-case VaR, worst-case CVaR, worst-case RVaR, and worst-case spectral risk measure, by presenting a unified framework. Furthermore, the compact solutions enhance tractability in optimization problems involving these risk measures, particularly when the true underlying distribution is unknown, and the first two moments are uncertain. The application of the extreme-case DRMs is illustrated with real data sets through numerical examples.

Suggested Citation

  • Shao, Hui & Zhang, Zhe George, 2023. "Distortion risk measure under parametric ambiguity," European Journal of Operational Research, Elsevier, vol. 311(3), pages 1159-1172.
    • Handle: RePEc:eee:ejores:v:311:y:2023:i:3:p:1159-1172
    DOI: 10.1016/j.ejor.2023.05.025
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    5. Peng Liu & Steven Vanduffel & Yi Xia, 2025. "Robust distortion risk metrics and portfolio optimization," Papers 2511.08662, arXiv.org.
    6. Boonen, Tim J. & Chen, Yuyu & Han, Xia & Wang, Qiuqi, 2025. "Optimal insurance design with Lambda-Value-at-Risk," European Journal of Operational Research, Elsevier, vol. 327(1), pages 232-246.
    7. Mengshuo Zhao & Narayanaswamy Balakrishnan & Chuancun Yin & Hui Shao, 2024. "Extremal cases of distortion risk measures with partial information," Papers 2404.13637, arXiv.org, revised Jul 2026.
    8. Mengshuo Zhao & Chuancun Yin, 2024. "Best- and worst-case Scenarios for GlueVaR distortion risk measure with Incomplete information," Papers 2409.19902, arXiv.org.
    9. Xiangyu Han & Yijun Hu & Ran Wang & Linxiao Wei, 2025. "On data-driven robust distortion risk measures for non-negative risks with partial information," Papers 2508.10682, arXiv.org.
    10. Marcelo Righi & Fernanda Muller, 2024. "A note on robust convex risk measures," Papers 2406.12999, arXiv.org, revised Jul 2025.
    11. Zuo, Baishuai & Yin, Chuancun, 2025. "Worst-case distortion riskmetrics and weighted entropy with partial information," European Journal of Operational Research, Elsevier, vol. 321(2), pages 476-492.
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