Asymptotic Equivalence and Adaptive Estimation for Robust Nonparametric Regression
The Annals of Statistics 37 , 3204-3235, (2009).
Asymptotic Equivalence and Adaptive Estimation for Robust Nonparametric Regression
The Annals of Statistics 37 , 3204-3235, (2009).
- 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.
- Paper: pdf file.
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