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Optimal global rates of convergence for noiseless regression estimation problems with adaptively chosen design

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  • Kohler, Michael

Abstract

Given the values of a measurable function m:Rd→R at n arbitrarily chosen points in Rd the problem of estimating m on whole Rd is considered. Here the estimate has to be defined such that the L1 error of the estimate (with integration with respect to a fixed but unknown probability measure) is small. Under the assumption that m is (p,C)-smooth (i.e., roughly speaking, m is p-times continuously differentiable) it is shown that the optimal minimax rate of convergence of the L1 error is n−p/d, where the upper bound is valid even if the support of the design measure is unbounded but the design measure satisfies some moment condition. Furthermore it is shown that this rate of convergence cannot be improved even if the function is not allowed to change with the size of the data.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:132:y:2014:i:c:p:197-208
    DOI: 10.1016/j.jmva.2014.08.008
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Tina Felber & Michael Kohler & Adam Krzyżak, 2015. "Adaptive density estimation based on real and artificial data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(1), pages 1-18, March.
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    Citations

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    Cited by:

    1. 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.
    2. Langer, Sophie, 2021. "Approximating smooth functions by deep neural networks with sigmoid activation function," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    3. 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.
    4. Michael Kohler & Adam Krzyżak & Reinhard Tent & Harro Walk, 2018. "Nonparametric quantile estimation using importance sampling," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(2), pages 439-465, April.

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