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August 2003 Nonparametric estimators which can be "plugged-in"
Peter J. Bickel, Ya'acov Ritov
Ann. Statist. 31(4): 1033-1053 (August 2003). DOI: 10.1214/aos/1059655904


We consider nonparametric estimation of an object such as a probability density or a regression function. Can such an estimator achieve the ratewise minimax rate of convergence on suitable function spaces, while, at the same time, when "plugged-in," estimate efficiently (at a rate of~$n^{-1/2}$ with the best constant) many functionals of the object? For example, can we have a density estimator whose definite integrals are efficient estimators of the cumulative distribution function? We show that this is impossible for very large sets, for example, expectations of all functions bounded by $M<\infty$. However, we also show that it is possible for sets as large as indicators of all quadrants, that is, distribution functions. We give appropriate constructions of such estimates.


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Peter J. Bickel. Ya'acov Ritov. "Nonparametric estimators which can be "plugged-in"." Ann. Statist. 31 (4) 1033 - 1053, August 2003.


Published: August 2003
First available in Project Euclid: 31 July 2003

zbMATH: 1058.62031
MathSciNet: MR2001641
Digital Object Identifier: 10.1214/aos/1059655904

Primary: 62F12 , 62G07 , 62G30

Keywords: Density estimation , Efficient estimator , Nonparametric regression

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 4 • August 2003
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