ESTIMATING NONLINEAR FUNCTIONS OF THE SPECTRAL DENSITY, USING A DATA-TAPER

Let f(omega) be the spectral density of a Gaussian stationary process. Consider periodogram-based estimators of integrals of certain non-lin ear functions zeta of f(omega), like H-T := integral(-pi)(pi) Lambda(o mega)zeta(I-T(omega))d omega, where Lambda(omega) is a bounded functio n of bounded variation, possibly depending on the sample size T. Then it is known that, under mild conditions on zeta, a central limit theor em holds for these statistics H-T if the non-tapered periodogram I-T(o mega) is used. In particular, Taniguchi (1980, J. Appl. Probab., 17, 7 3-83) gave a consistent and asymptotic normal estimator of integral(-p i)(pi) Lambda(omega)Phi(f(omega))d omega, choosing zeta to be a suitab le transform of a given function Phi. In this work we shall generalize this result to statistics H-T where a taper-modified periodogram is u sed. We apply our result to the use of data-tapers in nonparametric pe ak-insensitive spectrum estimation. This was introduced in von Sachs ( 1994, J. Time Ser. Anal., 15, 429-452) where the performance of this e stimator was shown to be substantially improved by using a taper.