Bayesian Decision Theoretic Scale-Adaptive Estimation of a Log-Spectral Density

Abstract

The problem of estimating the log-spectrum of a stationary Gaussian time series by Bayesianly induced shrinkage of empirical wavelet coefficients is studied. A model in the wavelet domain that accounts for distributional properties of the log-periodogram at levels of fine detail and approximate normality at coarse levels in the wavelet decomposition, is proposed. The smoothing procedure, called BAMS-LP (Bayesian Adaptive Multiscale Shrinker of Log-Periodogram), ensures that the reconstructed log-spectrum is as noise-free as possible. It is also shown that the resulting Bayes estimators are asymptotically optimal (in the frequentist sense). Comparisons with non-wavelet and wavelet-non-Bayesian methods are discussed.