A wavelet-Fisz approach to spectrum estimation

We propose a new approach to wavelet threshold estimation of spectral densities of stationary time series. Our proposal addresses the problem of heteroscedasticity and non-normality of the (tapered) periodogram. We estimate thresholds for the empirical wavelet coefficients of the periodogram as appropriate linear combinations of the periodogram values similar to empirical scaling coefficients. Our solution introduces 'asymptotically noise-free reconstruction thresholds' which parallels classical wavelet theory for nonparametric regression with independently and identically distributed Gaussian errors. Our simulations show promising results that clearly improve on existing approaches. In addition, we derive theoretical results on the near-optimal rate of convergence of the minimax mean-square risk for a class of spectral densities, including those of low regularity. Copyright 2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd