Technical Report: A wavelet-Fisz approach to spectrum estimation

We suggest a new approach to wavelet threshold estimation of spectral densities of stationary time series. It is well known that choosing appropriate thresholds to smooth the periodogram is difficult because non-parametric spectral estimation suffers from problems similar to curve estimation with a highly heteroscedastic and non-Gaussian error structure. Possible solutions that have been proposed are plug-in estimation of the variance of the empirical wavelet coefficients or the log-transformation of the periodogram. In this paper we propose an alternative method to address the problem of heteroscedasticity and non-normality. We estimate thresholds for the empirical wavelet coefficients of the (tapered) periodogram as appropriate linear combinations of the periodogram values similar to empirical scaling coefficients. Our solution permits the design of “asymptotically noise-free reconstruction thresholds”, paralleling classical wavelet theory for nonparametric regression with Gaussian white noise errors. Our simulation studies show promising results that clearly improve the classical approaches mentioned above. In addition, we derive theoretical results on the nearoptimal rate of convergence of the minimax mean-square risk for a class of spectral densities, including those of very low regularity.