WAVELET THRESHOLD ESTIMATORS FOR DATA WITH CORRELATED NOISE

Wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level-dependent soft threshold to the c oefficients in the wavelet transform. A variety of threshold choices i s proposed, including one based on an unbiased estimate of mean-square d error. The practical performance of the method is demonstrated on ex amples, including data from a neurophysiological context. The theoreti cal properties of the estimators are investigated by comparing them wi th an ideal but unattainable 'bench-mark', that can be considered in t he wavelet context as the risk obtained by ideal spatial adaptivity, a nd more generally is obtained by the use of an 'oracle' that provides information that is not actually available in the data. It is shown th at the level-dependent threshold estimator performs well relative to t he bench-mark risk, and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator. The wavelet domain st ructure of both short- and long-range dependent noise is considered, a nd in both cases it is shown that the estimators have near optimal beh aviour simultaneously in a wide range of function classes, adapting au tomatically 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.