ASYMPTOTIC EQUIVALENCE AND ADAPTIVE ESTIMATION FOR ROBUST NONPARAMETRIC REGRESSION

被引:11
作者
Cai, T. Tony [1 ]
Zhou, Harrison H. [2 ]
机构
[1] Univ Penn, Wharton Sch, Dept Stat, Philadelphia, PA 19104 USA
[2] Yale Univ, Dept Stat, New Haven, CT 06511 USA
基金
美国国家科学基金会;
关键词
Adaptivity; asymptotic equivalence; James-Stein estimator; moderate deviation; nonparametric regression; quantile coupling; robust estimation; unbounded loss function; wavelets; DENSITY-ESTIMATION; APPROXIMATION; INEQUALITY; MODELS; GARCH; SHARP;
D O I
10.1214/08-AOS681
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
070103 [概率论与数理统计]; 140311 [社会设计与社会创新];
摘要
Asymptotic equivalence theory developed in the literature so far are only for bounded loss functions. This limits the potential applications of the theory because many commonly used loss functions in statistical inference are unbounded. In this paper we develop asymptotic equivalence results for robust nonparametric regression with unbounded loss functions. The results imply that all the Gaussian nonparametric regression procedures can be robustified in a unified way. A key step in our equivalence argument is to bin the data and then take the median of each bin. The asymptotic equivalence results have significant practical implications. To illustrate the general principles of the equivalence argument we consider two important nonparametric inference problems: robust estimation of the reagression function and the estimation of a quadratic functional. In both cases easily implementable procedures are constructed and are shown to enjoy simultaneously a high degree of robustness and adaptivity. Other problems such as construction of confidence sets and nonparametric hypothesis testing can be handled in a similar fashion.
引用
收藏
页码:3204 / 3235
页数:32
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