Asymptotic Equivalence and Adaptive Estimation for Robust Nonparametric Regression
Tony Cai and Harrison Zhou
Abstract:
Asymptotic equivalence results 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. Through binning and
taking the medians of the binned data, the general nonparametric regression
model is turned into a standard Gaussian regression model, and then in
principle any procedures for Gaussian nonparametric regression can be
applied.
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
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.