Optimal global rates of convergence for nonparametric regression with unbounded data |
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Authors: | Michael Kohler Adam Krzyżak Harro Walk |
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Affiliation: | 1. Department of Mathematics, Technische Universität Darmstadt, Schlossgartenstr. 7, D-64289 Darmstadt, Germany;2. Department of Computer Science and Software Engineering, Concordia University, 1455 De Maisonneuve Blvd. West, Montreal, Quebec, Canada H3G 1M8;3. Department of Mathematics, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany |
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Abstract: | Estimation of regression functions from independent and identically distributed data is considered. The L2 error with integration with respect to the design measure is used as an error criterion. Usually in the analysis of the rate of convergence of estimates a boundedness assumption on the explanatory variable X is made besides smoothness assumptions on the regression function and moment conditions on the response variable Y. In this article we consider the kernel estimate and show that by replacing the boundedness assumption on X by a proper moment condition the same (optimal) rate of convergence can be shown as for bounded data. This answers Question 1 in Stone [1982. Optimal global rates of convergence for nonparametric regression. Ann. Statist., 10, 1040–1053]. |
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Keywords: | Regression Kernel estimate Rate of convergence |
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