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A wavelet-Fisz approach to spectrum estimation
by Piotr Fryzlewicz, Guy Nason and Rainer von Sachs
We propose a new approach to wavelet threshold estimation of spectral densities of station-
ary 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 thresh-
olds” which parallels classical wavelet theory for nonparametric regression with iid 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.
Key words: spectral density estimation, wavelet thresholding, wavelet-Fisz, periodogram,
Besov spaces, smoothing.
Full text of the paper (pdf), which appeared in the Journal of Time Series Analysis, 2008
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