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Bickel, Peter J.; Ritov, Ya’acov

Nonparametric estimators which can be “plugged-in”. (English) Zbl 1058.62031

Ann. Stat. 31, No. 4, 1033-1053 (2003).
Summary: 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.

Cited in 1 Review
Cited in 37 Documents

MSC:

62G07 Density estimation
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62C20 Minimax procedures in statistical decision theory
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References:

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[18] BERKELEY, CALIFORNIA 94720-3860 E-MAIL: bickel@stat.berkeley.edu DEPARTMENT OF STATISTICS HEBREW UNIVERSITY JERUSALEM 91905 ISRAEL E-MAIL: yaacov@mscc.huji.ac.il
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