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Nonparametric estimation of a function from noiseless observations at random points

Author

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  • Bauer, Benedikt
  • Devroye, Luc
  • Kohler, Michael
  • Krzyżak, Adam
  • Walk, Harro

Abstract

In this paper we study the problem of estimating a function from n noiseless observations of function values at randomly chosen points. These points are independent copies of a random variable whose density is bounded away from zero on the unit cube and vanishes outside. The function to be estimated is assumed to be (p,C)-smooth, i.e., (roughly speaking) it is p times continuously differentiable. Our main results are that the supremum norm error of a suitably defined spline estimate is bounded in probability by {ln(n)∕n}p∕d for arbitrary p and d and that this rate of convergence is optimal in minimax sense.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:160:y:2017:i:c:p:93-104
    DOI: 10.1016/j.jmva.2017.05.010
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    References listed on IDEAS

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    1. Joldes, Grand Roman & Chowdhury, Habibullah Amin & Wittek, Adam & Doyle, Barry & Miller, Karol, 2015. "Modified moving least squares with polynomial bases for scattered data approximation," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 893-902.
    2. 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.
    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.
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    Citations

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    Cited by:

    1. Michael Kohler & Adam Krzyżak, 2020. "Estimating quantiles in imperfect simulation models using conditional density estimation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(1), pages 123-155, February.
    2. Christian Bender & Nikolaus Schweizer, 2019. "`Regression Anytime' with Brute-Force SVD Truncation," Papers 1908.08264, arXiv.org, revised Oct 2020.

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