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An Upper Bound of the Bias of Nadaraya-Watson Kernel Regression under Lipschitz Assumptions

Author

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  • Samuele Tosatto

    (Computer Science Department, Technische Universität Darmstadt, 64289 Darmstadt, Germany)

  • Riad Akrour

    (Computer Science Department, Technische Universität Darmstadt, 64289 Darmstadt, Germany)

  • Jan Peters

    (Computer Science Department, Technische Universität Darmstadt, 64289 Darmstadt, Germany
    Computer Science Department, Max Planck Institute for Intelligent Systems, 70569 Stuttgart, Germany)

Abstract

The Nadaraya-Watson kernel estimator is among the most popular nonparameteric regression technique thanks to its simplicity. Its asymptotic bias has been studied by Rosenblatt in 1969 and has been reported in several related literature. However, given its asymptotic nature, it gives no access to a hard bound. The increasing popularity of predictive tools for automated decision-making surges the need for hard (non-probabilistic) guarantees. To alleviate this issue, we propose an upper bound of the bias which holds for finite bandwidths using Lipschitz assumptions and mitigating some of the prerequisites of Rosenblatt’s analysis. Our bound has potential applications in fields like surgical robots or self-driving cars, where some hard guarantees on the prediction-error are needed.

Suggested Citation

  • Samuele Tosatto & Riad Akrour & Jan Peters, 2020. "An Upper Bound of the Bias of Nadaraya-Watson Kernel Regression under Lipschitz Assumptions," Stats, MDPI, vol. 4(1), pages 1-17, December.
    • Handle: RePEc:gam:jstats:v:4:y:2020:i:1:p:1-17:d:470489
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    References listed on IDEAS

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    1. Max Köhler & Anja Schindler & Stefan Sperlich, 2014. "A Review and Comparison of Bandwidth Selection Methods for Kernel Regression," International Statistical Review, International Statistical Institute, vol. 82(2), pages 243-274, August.
    2. Mack, Y. P. & Müller, Hans-Georg, 1988. "Convolution type estimators for nonparametric regression," Statistics & Probability Letters, Elsevier, vol. 7(3), pages 229-239, December.
    3. Bansal, Ravi & Gallant, A. Ronald & Hussey, Robert & Tauchen, George, 1995. "Nonparametric estimation of structural models for high-frequency currency market data," Journal of Econometrics, Elsevier, vol. 66(1-2), pages 251-287.
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