Wavelets and variance reduction in non-parametric transfer function estimation
Sippe G. Douma, Thomas J. de Hoog, Paul M.J. Van den Hof · 2002
A variance reduction scheme is presented for non-parametric transfer function estimators based on the use of wavelets as an alternative to the traditional spectral windowing. The latter can be generalized into a variance reduction method based on thresholding (omitting or altering) the coefficients of an orthogonal series expansion of the estimator to be smoothed. The choice of threshold level is crucial, distinguishing between coefficients related predominantly to estimation errors and those associated with the underlying true function. The standard wavelet threshold operation with a constant or level-dependent threshold can not be applied to wavelet coefficients of spectral density functions. The nonstationarity in the statistical properties of these estimators reveals itself in the wavelet domain as significant peaks. An efficient threshold level should follow the standard deviation of each wavelet coefficient. New exact expressions of the standard deviation are presented. An estimator based on these expressions is proved to provide an appropriate threshold level.