Nonparametric estimators which can be plugged-in

• 成果类型: Article
• 署名作者: Bickel, PJ; Ritov, Y
• 署名单位: University of California System; University of California Berkeley; Hebrew University of Jerusalem
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 2003
• 页码: 1033-1053
• 关键词: large-sample; empirical distributions; asymptotic equivalence; DENSITY-ESTIMATION; regression; selection

摘要

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 < infinity. 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.