Robust Nonparametric Estimation via Wavelet Median Regression
Tony Cai and Harrison Zhou
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Abstract:
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.
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- Other related papers:
Cai, T. (1999).
Adaptive wavelet estimation: a block thresholding and oracle inequality approach.
The Annals of Statistics 27, 898-924.Cai, T. (2002).
On block thresholding in wavelet regression: Adaptivity, block size, and threshold level.
Statistica Sinica 12, 1241-1273.Cai, T. & Low, M. (2005).
Nonparametric function estimation over shrinking neighborhoods: Superefficiency and adaptation.
The Annals of Statistics 33, 184-213.Cai, T. & Brown, L.D. (1998).
Wavelet shrinkage for nonequispaced samples.
The Annals of Statistics 26, 1783-1799.