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Quantitative robustness of instance ranking problems

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  • Tino Werner

    (Institute for Mathematics, Carl von Ossietzky University Oldenburg)

Abstract

Instance ranking problems intend to recover the ordering of the instances in a data set with applications in scientific, social and financial contexts. In this work, we concentrate on the global robustness of parametric instance ranking problems in terms of the breakdown point which measures the fraction of samples that need to be perturbed in order to let the estimator take unreasonable values. Existing breakdown point notions do not cover ranking problems so far. We propose to define a breakdown of the estimator as a sign-reversal of all components which causes the predicted ranking to be potentially completely inverted; therefore, we call it the order-inversal breakdown point (OIBDP). We will study the OIBDP, based on a linear model, for several different carefully distinguished ranking problems and provide least favorable outlier configurations, characterizations of the order-inversal breakdown point and sharp asymptotic upper bounds. We also compute empirical OIBDPs.

Suggested Citation

  • Tino Werner, 2023. "Quantitative robustness of instance ranking problems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(2), pages 335-368, April.
    • Handle: RePEc:spr:aistmt:v:75:y:2023:i:2:d:10.1007_s10463-022-00847-1
    DOI: 10.1007/s10463-022-00847-1
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    References listed on IDEAS

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    1. Tino Werner, 2022. "Elicitability of Instance and Object Ranking," Decision Analysis, INFORMS, vol. 19(2), pages 123-140, June.
    2. Hema Yoganarasimhan, 2020. "Search Personalization Using Machine Learning," Management Science, INFORMS, vol. 66(3), pages 1045-1070, March.
    3. Hubert, Mia, 1997. "The breakdown value of the L1 estimator in contingency tables," Statistics & Probability Letters, Elsevier, vol. 33(4), pages 419-425, May.
    4. Shinichi Sakata & Halbert White, 1998. "High Breakdown Point Conditional Dispersion Estimation with Application to S&P 500 Daily Returns Volatility," Econometrica, Econometric Society, vol. 66(3), pages 529-568, May.
    5. Peter Ruckdeschel & Nataliya Horbenko, 2012. "Yet another breakdown point notion: EFSBP," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(8), pages 1025-1047, November.
    6. Marc G. Genton & André Lucas, 2003. "Comprehensive definitions of breakdown points for independent and dependent observations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(1), pages 81-94, February.
    7. Leon Yang Chu & Hamid Nazerzadeh & Heng Zhang, 2020. "Position Ranking and Auctions for Online Marketplaces," Management Science, INFORMS, vol. 66(8), pages 3617-3634, August.
    8. Hennig, Christian, 2008. "Dissolution point and isolation robustness: Robustness criteria for general cluster analysis methods," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1154-1176, July.
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