Fryzlewicz, Piotr, Nason, Guy P. and von Sachs, Rainer
(2008)
A wavelet-Fisz approach to spectrum estimation.
Journal of time series analysis, 29
(5).
pp. 868-880.
ISSN 0143-9782
Abstract
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 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 near-optimal rate of convergence
of the minimax mean-square risk for a class of spectral densities, including those of very
low regularity.
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