Authors: Johnstone, Iain M.¹; Silverman, Bernard W.²

Source: Journal of the Royal Statistical Society: Series B (Statistical Methodology), Volume 59, Number 2, 1997 , pp. 319-351(33)

Publisher: Wiley-Blackwell

Abstract:

Wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level-dependent soft threshold to the coefficients in the wavelet transform. A variety of threshold choices is proposed, including one based on an unbiased estimate of mean-squared error. The practical performance of the method is demonstrated on examples, including data from a neurophysiological context. The theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable `bench-mark', that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity, and more generally is obtained by the use of an `oracle' that provides information that is not actually available in the data. It is shown that the level-dependent threshold estimator performs well relative to the bench-mark risk, and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator. The wavelet domain structure of both short- and long-range dependent noise is considered, and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes, adapting automatically to the regularity properties of the underlying model. The proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem.

Keywords: adaptive estimation; decision theory; ion channels; level-dependent thresholding; long-range dependence; minimax estimation; non-linear estimators; nonparametric regression; oracle inequality; wavelet transform

Document Type: Original article

DOI: http://dx.doi.org/10.1111/1467-9868.00071

Affiliations: 1: Stanford University, US, 2: Bristol University, UK

Publication date: 1997-01-01

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