Bayesian Decision Theoretic Scale-Adaptive Estimation of a Log-Spectral Density
Marianna Pensky, Brani Vidaković · 2003
Abstract: The problem of estimating the log-spectrum of a stationary Gaussiantime series by Bayesianly induced shrinkage of empirical wavelet coefficients is studied. A model in the wavelet domain that accounts for distributional propertiesof the log-periodogram at levels of fine detail and approximate normality at coarse levels in the wavelet decomposition, is proposed. The smoothing procedure, calledBAMS-LP (Bayesian Adaptive Multiscale Shrinker of Log-Periodogram), ensures that the reconstructed log-spectrum is as noise-free as possible. It is also shown thatthe resulting Bayes estimators are asymptotically optimal (in the frequentist sense). Comparisons with non-wavelet and wavelet-non-Bayesian methods are discussed. Key words and phrases: Spectral Density, Log-Spectral Density, Wavelets. 1 Introduction Any statistical inference in time series can be conducted in time and frequency domains. Themethods are complementary and provide different insights. Spectral analysis, and in particular, estimation of spectral density is an indispensable tool for exploring the frequency behavior of atime series. Wavelet shrinkage methods have successfully been applied to the spectral density esti-mation in work of Lumeau et al. (1993), Moulin (1992, 1994), Gao (1992, 1993a,b) from the classical view-point. In this paper we propose a novel wavelet-shrinkage method, based onintrinsic shrinkage property of Bayes rules. The proposed shrinkage rules resulting from hierarchical Bayes statistical models are both realistic, i.e., describe data accurately, and capable ofincorporating the available prior information on smoothness of functions represented by their wavelet coefficients.Let {