The Annals of Statistics

Robust nonparametric estimation via wavelet median regression

Lawrence D. Brown, T. Tony Cai, and Harrison H. Zhou

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In this paper we develop a nonparametric regression method that is simultaneously adaptive over a wide range of function classes for the regression function and robust over a large collection of error distributions, including those that are heavy-tailed, and may not even possess variances or means. Our approach is to first use local medians to turn the problem of nonparametric regression with unknown noise distribution into a standard Gaussian regression problem and then apply a wavelet block thresholding procedure to construct an estimator of the regression function. It is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the Besov classes, without prior knowledge of the smoothness of the underlying functions or prior knowledge of the error distribution. The estimator also automatically adapts to the local smoothness of the underlying function, and attains the local adaptive minimax rate for estimating functions at a point.

A key technical result in our development is a quantile coupling theorem which gives a tight bound for the quantile coupling between the sample medians and a normal variable. This median coupling inequality may be of independent interest.

Article information

Ann. Statist., Volume 36, Number 5 (2008), 2055-2084.

First available in Project Euclid: 13 October 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties

Adaptivity asymptotic equivalence James–Stein estimator moderate large deviation nonparametric regression quantile coupling robust estimation wavelets


Brown, Lawrence D.; Cai, T. Tony; Zhou, Harrison H. Robust nonparametric estimation via wavelet median regression. Ann. Statist. 36 (2008), no. 5, 2055--2084. doi:10.1214/07-AOS513.

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