A wavelet-Fisz approach to spectrum estimation

Piotr Fryzlewicz, Guy P. Nason and Rainer von Sachs

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

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.

Keywords: spectral density estimation; wavelet thresholding; wavelet-Fisz; periodogram; Besov spaces; smoothing (search for similar items in EconPapers)
JEL-codes: C1 (search for similar items in EconPapers)
Date: 2008-09
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Citations: View citations in EconPapers (4)

Published in Journal of Time Series Analysis, September, 2008, 29(5), pp. 868-880. ISSN: 0143-9782

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Persistent link: https://EconPapers.repec.org/RePEc:ehl:lserod:25186

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