The Annals of Statistics

Nonparametric estimators which can be "plugged-in"

Peter J. Bickel and Ya'acov Ritov

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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.

Article information

Ann. Statist., Volume 31, Number 4 (2003), 1033-1053.

First available in Project Euclid: 31 July 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62G30: Order statistics; empirical distribution functions 62F12: Asymptotic properties of estimators

Efficient estimator density estimation nonparametric regression


Bickel, Peter J.; Ritov, Ya'acov. Nonparametric estimators which can be "plugged-in". Ann. Statist. 31 (2003), no. 4, 1033--1053. doi:10.1214/aos/1059655904.

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