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Wavelet Threshold Estimators for Data with Correlated Noise
Author(s) -
Johnstone Iain M.,
Silverman Bernard W.
Publication year - 1997
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/1467-9868.00071
Subject(s) - estimator , minimax , wavelet , context (archaeology) , mathematics , noise (video) , mean squared error , range (aeronautics) , computer science , algorithm , statistics , mathematical optimization , artificial intelligence , paleontology , materials science , composite material , image (mathematics) , biology
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.