Asymptotic equivalence and adaptive estimation for robust nonparametric regression

TL;DR

This paper extends asymptotic equivalence theory to unbounded loss functions in robust nonparametric regression, enabling the development of robust, adaptive procedures for estimation and inference with practical significance.

Contribution

It introduces asymptotic equivalence results for unbounded loss functions, allowing Gaussian procedures to be robustified and applied to various nonparametric inference problems.

Findings

Robust nonparametric regression procedures are developed and shown to be both robust and adaptive.

The median-of-bins technique is key to establishing asymptotic equivalence.

Procedures for confidence sets and hypothesis testing are also adaptable.

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.