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On the rate of convergence of the functional k-nearest neighbor estimates

Abstract : Let F be a separable Banach space, and let (X, Y) be a random pair taking values in F x R. Motivated by a broad range of potential applications, we investigate rates of convergence of the k-nearest neighbor estimate r_n(x) of the regression function r(x) = E[Y|X = x], based on n independent copies of the pair (X, Y). Using compact embedding theory, we present explicit and general finite sample bounds on the expected squared difference E[(r_n(X) - r(X)]^2], and particularize our results to classical function spaces such as Sobolev spaces, Besov spaces, and reproducing kernel Hilbert spaces.
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Submitted on : Sunday, December 1, 2013 - 1:07:12 PM
Last modification on : Saturday, March 28, 2020 - 2:20:59 AM

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Gérard Biau, Frédéric Cérou, Arnaud Guyader. On the rate of convergence of the functional k-nearest neighbor estimates. IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2010, IT-56 (4), pp.2034-2040. ⟨10.1109/TIT.2010.2040857⟩. ⟨hal-00911993⟩

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