Wavelet Threshold Estimators for Data with Correlated Noise

Wavelet threshold estimators for data with stationary correlated noise are constructed by the following prescription. First, form the discrete wavelet transform of the data points. Next, apply a level-dependent soft threshold to the individual coeecients, allowing the thresholds to depend on the level in the wavelet transform. Finally, transform back to obtain the estimate in the original domain. The threshold used at level j is s j p 2 log n, where s j is the standard deviation of the coeecients at that level, and n is the overall sample size. The minimax properties of the estimators are investigated by considering a general problem in multivariate normal decision theory, concerned with the estimation of the mean vector of a general multivariate normal distribution subject to squared error loss. An ideal risk is obtained by the use of anòracle' that provides the optimum diagonal projection estimate. This`benchmark' risk can be considered in its own right as a measure of the sparseness of the signal relative to the noise process, and in the wavelet context it can be considered as the risk obtained by ideal spatial adaptivity. It is shown that the level-dependent threshold estimator performs well relative to the benchmark risk, and that its minimax behaviour cannot be improved upon in order of magnitude by any other estimator.