Statistics Papers

Document Type

Journal Article

Date of this Version

2009

Publication Source

The Annals of Statistics

Volume

37

Issue

6A

Start Page

3204

Last Page

3235

DOI

10.1214/08-AOS681

Abstract

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.

Keywords

adaptivity, asymptotic equivalence, James–Stein estimator, moderate deviation, nonparametric regression, quantile coupling, robust estimation, unbounded loss function, wavelets

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Date Posted: 27 November 2017

This document has been peer reviewed.