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June, 1990 A Risk Bound in Sobolev Class Regression
Grigori K. Golubev, Michael Nussbaum
Ann. Statist. 18(2): 758-778 (June, 1990). DOI: 10.1214/aos/1176347624


For nonparametric regression estimation, when the unknown function belongs to a Sobolev smoothness class, sharp risk bounds for integrated mean square error have been found recently which improve on optimal rates of convergence results. The key to these has been the fact that under normality of the errors, the minimax linear estimator is asymptotically minimax in the class of all estimators. We extend this result to the nonnormal case, when the noise distribution is unknown. The pertaining lower asymptotic risk bound is established, based on an analogy with a location model in the independent identically distributed case. Attainment of the bound and its relation to adaptive optimal smoothing are discussed.


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Grigori K. Golubev. Michael Nussbaum. "A Risk Bound in Sobolev Class Regression." Ann. Statist. 18 (2) 758 - 778, June, 1990.


Published: June, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0713.62047
MathSciNet: MR1056335
Digital Object Identifier: 10.1214/aos/1176347624

Primary: 62G20
Secondary: 62C20, 62G05

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 2 • June, 1990
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