Estimating non-linear functions of the spectral density,using a data-taper

Letf() be the spectral density of a Gaussian stationary process. Consider periodogram-based estimators of integrals of certain non-linear functions off(), like, where () is a bounded function of bounded variation, possibly depending on the sample sizeT. Then it is known that, under mild conditions on , a central limit theorem holds for these statisticsH_T if the non-tapered periodogramI_T() is used. In particular, Taniguchi (1980,J. Appl. Probab.,17, 73–83) gave a consistent and asymptotic normal estimator of, choosing to be a suitable transform of a given function . In this work we shall generalize this result to statisticsH_T where a taper-modified periodogram is used. We apply our result to the use of data-tapers in nonparametric peak-insensitive spectrum estimation. This was introduced in von Sachs (1994,J. Time Ser. Anal.,15, 429–452) where the performance of this estimator was shown to be substantially improved by using a taper.This work has been supported by the Deutsche Forschungsgemeinschaft.