Open Access
December 2009 Asymptotic equivalence and adaptive estimation for robust nonparametric regression
T. Tony Cai, Harrison H. Zhou
Ann. Statist. 37(6A): 3204-3235 (December 2009). DOI: 10.1214/08-AOS681


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 regression 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.


Download Citation

T. Tony Cai. Harrison H. Zhou. "Asymptotic equivalence and adaptive estimation for robust nonparametric regression." Ann. Statist. 37 (6A) 3204 - 3235, December 2009.


Published: December 2009
First available in Project Euclid: 17 August 2009

zbMATH: 1191.62070
MathSciNet: MR2549558
Digital Object Identifier: 10.1214/08-AOS681

Primary: 62G08
Secondary: 62G20

Keywords: Adaptivity , ‎asymptotic ‎equivalence , James–Stein estimator , Moderate deviation , Nonparametric regression , Quantile coupling , robust estimation , unbounded loss function , Wavelets

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 6A • December 2009
Back to Top