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

Abstract

Letf(ω) be the spectral density of a Gaussian stationary process. Consider periodogram-based estimators of integrals of certain non-linear functions ζ off(ω), like$H_T : = \smallint _{ - \pi }^\pi \Lambda (\omega )\zeta \left( {I_T \left( \omega \right)} \right)d\omega$, where Λ(ω) is a bounded function of bounded variation, possibly depending on the sample sizeT. Then it is known that, under mild conditions on ζ, a central limit theorem holds for these statisticsH _T if the non-tapered periodogramI _T(ω) is used. In particular, Taniguchi (1980,J. Appl. Probab.,17, 73–83) gave a consistent and asymptotic normal estimator of$\smallint _{ - \pi }^\pi \Lambda (\omega )\Phi \left( {f\left( \omega \right)} \right)d\omega$, choosing ζ to be a suitable transform of a given function Φ. 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.