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Domains of weak continuity of statistical functionals with a view toward robust statistics

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  • Krätschmer, Volker
  • Schied, Alexander
  • Zähle, Henryk

Abstract

Many standard estimators such as several maximum likelihood estimators or the empirical estimator for any law-invariant convex risk measure are not (qualitatively) robust in the classical sense. However, these estimators may nevertheless satisfy a weak robustness property (Krätschmer et al. (2012, 2014)) or a local robustness property (Zähle (2016)) on relevant sets of distributions. One aim of our paper is to identify sets of local robustness, and to explain the benefit of the knowledge of such sets. For instance, we will be able to demonstrate that many maximum likelihood estimators are robust on their natural parametric domains. A second aim consists in extending the general theory of robust estimation to our local framework. In particular we provide a corresponding Hampel-type theorem linking local robustness of a plug-in estimator with a certain continuity condition.

Suggested Citation

  • Krätschmer, Volker & Schied, Alexander & Zähle, Henryk, 2017. "Domains of weak continuity of statistical functionals with a view toward robust statistics," Journal of Multivariate Analysis, Elsevier, vol. 158(C), pages 1-19.
    • Handle: RePEc:eee:jmvana:v:158:y:2017:i:c:p:1-19
    DOI: 10.1016/j.jmva.2017.02.005
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    References listed on IDEAS

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    1. 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.
    2. Paul Embrechts & Bin Wang & Ruodu Wang, 2015. "Aggregation-robustness and model uncertainty of regulatory risk measures," Finance and Stochastics, Springer, vol. 19(4), pages 763-790, October.
    3. Volker Krätschmer & Alexander Schied & Henryk Zähle, 2014. "Comparative and qualitative robustness for law-invariant risk measures," Finance and Stochastics, Springer, vol. 18(2), pages 271-295, April.
    4. Krätschmer, Volker & Schied, Alexander & Zähle, Henryk, 2012. "Qualitative and infinitesimal robustness of tail-dependent statistical functionals," Journal of Multivariate Analysis, Elsevier, vol. 103(1), pages 35-47, January.
    5. Andrzej Ruszczyński & Alexander Shapiro, 2006. "Optimization of Convex Risk Functions," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 433-452, August.
    6. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Post-Print hal-00413729, HAL.
    7. Volker Kratschmer & Alexander Schied & Henryk Zahle, 2012. "Comparative and qualitative robustness for law-invariant risk measures," Papers 1204.2458, arXiv.org, revised Jan 2014.
    8. Patrick Cheridito & Tianhui Li, 2009. "Risk Measures On Orlicz Hearts," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 189-214, April.
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    Citations

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    Cited by:

    1. Paul Embrechts & Alexander Schied & Ruodu Wang, 2018. "Robustness in the Optimization of Risk Measures," Papers 1809.09268, arXiv.org, revised Feb 2021.
    2. Henryk Zähle, 2022. "A concept of copula robustness and its applications in quantitative risk management," Finance and Stochastics, Springer, vol. 26(4), pages 825-875, October.
    3. Ruodu Wang & Johanna F. Ziegel, 2018. "Scenario-based Risk Evaluation," Papers 1808.07339, arXiv.org, revised May 2021.
    4. Patrick Kern & Axel Simroth & Henryk Zähle, 2020. "First-order sensitivity of the optimal value in a Markov decision model with respect to deviations in the transition probability function," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(1), pages 165-197, August.
    5. Ruodu Wang & Johanna F. Ziegel, 2021. "Scenario-based risk evaluation," Finance and Stochastics, Springer, vol. 25(4), pages 725-756, October.
    6. Niushan Gao & Foivos Xanthos, 2024. "A note on continuity and asymptotic consistency of measures of risk and variability," Papers 2405.09766, arXiv.org, revised Oct 2024.
    7. Sainan Zhang & Huifu Xu, 2022. "Insurance premium-based shortfall risk measure induced by cumulative prospect theory," Computational Management Science, Springer, vol. 19(4), pages 703-738, October.
    8. Jiang, Jie & Peng, Shen, 2024. "Mathematical programs with distributionally robust chance constraints: Statistical robustness, discretization and reformulation," European Journal of Operational Research, Elsevier, vol. 313(2), pages 616-627.

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