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Wavelet Threshold Estimators for Data with Correlated Noise

Iain M. Johnstone and Bernard W. Silverman
Journal of the Royal Statistical Society. Series B (Methodological)
Vol. 59, No. 2 (1997), pp. 319-351
Published by: Wiley for the Royal Statistical Society
Stable URL: http://www.jstor.org/stable/2346049
Page Count: 33
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Wavelet Threshold Estimators for Data with Correlated Noise
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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 `benchmark', 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.

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