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A bias bound approach to nonparametric inference

Author

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  • Susanne M. Schennach

    (Institute for Fiscal Studies and Brown University)

Abstract
The traditional approach to obtain valid con?dence intervals for nonparametric quantities is to select a smoothing parameter such that the bias of the estimator is negligible relative to its standard deviation. While this approach is apparently simple, it has two drawbacks: First, the question of optimal bandwidth selection is no longer well-de?ned, as it is not clear what ratio of bias to standard deviation should be considered negligible. Second, since the bandwidth choice necessarily deviates from the optimal (mean squares-minimizing) bandwidth, such a con?dence interval is very inefficient. To address these issues, we construct valid con?dence intervals that account for the presence of a nonnegligible bias and thus make it possible to perform inference with optimal mean squared error minimizing bandwidths. The key difficulty in achieving this involves ?nding a strict, yet feasible, bound on the bias of a nonparametric estimator. It is well-known that it is not possible to consistently estimate the point-wise bias of an optimal nonparametric estimator (for otherwise, one could subtract it and obtain a faster convergence rate violating Stone's bounds on optimal convergence rate). Nevertheless, we ?nd that, under minimal primitive assumptions, it is possible to consistently estimate an upper bound on the magnitude of the bias, which is su?cient to deliver a valid con?dence interval whose length decreases at the optimal rate and which does not contradict Stone’s results.

Suggested Citation

  • Susanne M. Schennach, 2015. "A bias bound approach to nonparametric inference," CeMMAP working papers CWP71/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    • Handle: RePEc:ifs:cemmap:71/15
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    References listed on IDEAS

    as
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    10. Politis, Dimitris N. & Romano, Joseph P., 1999. "Multivariate Density Estimation with General Flat-Top Kernels of Infinite Order," Journal of Multivariate Analysis, Elsevier, vol. 68(1), pages 1-25, January.
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    Cited by:

    1. Byunghoon Kang, 2017. "Inference in Nonparametric Series Estimation with Data-Dependent Undersmoothing," Working Papers 170712442, Lancaster University Management School, Economics Department.
    2. Kato, Kengo & Sasaki, Yuya, 2019. "Uniform confidence bands for nonparametric errors-in-variables regression," Journal of Econometrics, Elsevier, vol. 213(2), pages 516-555.
    3. Timothy B Armstrong & Michal Kolesár, 2018. "A Simple Adjustment for Bandwidth Snooping," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 85(2), pages 732-765.
    4. Byunghoon Kang, 2018. "Inference in Nonparametric Series Estimation with Specification Searches for the Number of Series Terms," Working Papers 240829404, Lancaster University Management School, Economics Department.
    5. Sebastian Calonico & Matias D. Cattaneo & Max H. Farrell, 2018. "Coverage Error Optimal Confidence Intervals for Local Polynomial Regression," Papers 1808.01398, arXiv.org, revised Jul 2021.
    6. Timothy B. Armstrong, 2018. "Adaptation Bounds for Confidence Bands under Self-Similarity," Cowles Foundation Discussion Papers 2146R, Cowles Foundation for Research in Economics, Yale University, revised Jul 2019.
    7. Timothy B. Armstrong & Michal Kolesár, 2018. "Optimal Inference in a Class of Regression Models," Econometrica, Econometric Society, vol. 86(2), pages 655-683, March.
    8. Sebastian Calonico & Matias D. Cattaneo & Max H. Farrell, 2018. "On the Effect of Bias Estimation on Coverage Accuracy in Nonparametric Inference," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(522), pages 767-779, April.
    9. Kato, Kengo & Sasaki, Yuya, 2018. "Uniform confidence bands in deconvolution with unknown error distribution," Journal of Econometrics, Elsevier, vol. 207(1), pages 129-161.
    10. Byunghoon Kang, 2019. "Inference in Nonparametric Series Estimation with Specification Searches for the Number of Series Terms," Papers 1909.12162, arXiv.org, revised Feb 2020.
    11. Kengo Kato & Yuya Sasaki & Takuya Ura, 2021. "Robust inference in deconvolution," Quantitative Economics, Econometric Society, vol. 12(1), pages 109-142, January.
    12. Yuya Sasaki & Takuya Ura, 2021. "Slow Movers in Panel Data," Papers 2110.12041, arXiv.org.
    13. Harold D. Chiang & Kengo Kato & Yuya Sasaki & Takuya Ura, 2021. "Linear programming approach to nonparametric inference under shape restrictions: with an application to regression kink designs," Papers 2102.06586, arXiv.org.

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