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Optimal global rates of convergence for interpolation problems with random design

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  • Kohler, Michael
  • Krzyżak, Adam

Abstract

Given a sample of a d-dimensional design variable X and observations of the corresponding values of a measurable function m:Rd→R without additional errors, we are interested in estimating m on whole Rd such that the L1 error (with integration with respect to the design measure) of the estimate is small. Under the assumption that the support of X is bounded and that m is (p,C)-smooth (i.e., roughly speaking, m is p-times continuously differentiable) we derive the minimax lower and upper bounds on the L1 error.

Suggested Citation

  • Kohler, Michael & Krzyżak, Adam, 2013. "Optimal global rates of convergence for interpolation problems with random design," Statistics & Probability Letters, Elsevier, vol. 83(8), pages 1871-1879.
    • Handle: RePEc:eee:stapro:v:83:y:2013:i:8:p:1871-1879
    DOI: 10.1016/j.spl.2013.04.018
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    References listed on IDEAS

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    1. Kohler, Michael & Krzyzak, Adam & Walk, Harro, 2006. "Rates of convergence for partitioning and nearest neighbor regression estimates with unbounded data," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 311-323, February.
    2. Liitiäinen, Elia & Corona, Francesco & Lendasse, Amaury, 2010. "Residual variance estimation using a nearest neighbor statistic," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 811-823, April.
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    Citations

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    Cited by:

    1. Leluc, Rémi & Portier, François & Zhuman, Aigerim & Segers, Johan, 2023. "Speeding up Monte Carlo Integration: Control Neighbors for Optimal Convergence," LIDAM Discussion Papers ISBA 2023019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Bauer, Benedikt & Devroye, Luc & Kohler, Michael & Krzyżak, Adam & Walk, Harro, 2017. "Nonparametric estimation of a function from noiseless observations at random points," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 93-104.
    3. Kohler, Michael, 2014. "Optimal global rates of convergence for noiseless regression estimation problems with adaptively chosen design," Journal of Multivariate Analysis, Elsevier, vol. 132(C), pages 197-208.
    4. Kohler, Michael & Krzyżak, Adam & Walk, Harro, 2014. "Nonparametric recursive quantile estimation," Statistics & Probability Letters, Elsevier, vol. 93(C), pages 102-107.
    5. Christian Bender & Nikolaus Schweizer, 2019. "`Regression Anytime' with Brute-Force SVD Truncation," Papers 1908.08264, arXiv.org, revised Oct 2020.
    6. Ann-Kathrin Bott & Tina Felber & Michael Kohler, 2015. "Estimation of a density in a simulation model," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(3), pages 271-285, September.
    7. Benedikt Bauer & Felix Heimrich & Michael Kohler & Adam Krzyżak, 2019. "On estimation of surrogate models for multivariate computer experiments," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(1), pages 107-136, February.

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