### ESTIMATING NON-LINEAR FUNCTIONS OF THE SPECTRAL DENSITY, USING A DATA-TAPER

#### RAINER VON SACHS

Sonderforschungsbereich 123, Universität Heidelberg and
Fachbereich Mathematik, Universität Kaiserslautern, Erwin-Schrödinger-Strasse,
67653 Kaiserslautern, Gebäude 48/516.1, Germany

(Received September 28, 1992; revised August 16, 1993)

Abstract.    Let f(omega) be the spectral density of a Gaussian stationary process. Consider periodogram-based estimators of integrals of certain non-linear functions zeta of f(omega), like HT := \int pi-pi Lambda(omega) zeta(IT(omega))domega, where Lambda(omega) is a bounded function of bounded variation, possibly depending on the sample size T. Then it is known that, under mild conditions on zeta, a central limit theorem holds for these statistics HT if the non-tapered periodogram IT(omega) is used. In particular, Taniguchi (1980, J. Appl. Probab., 17, 73-83) gave a consistent and asymptotic normal estimator of \int pi-piLambda(omega) Phi(f(omega))domega, choosing zeta to be a suitable transform of a given function Phi. In this work we shall generalize this result to statistics HT 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.

Key words and phrases:    Gaussian stationary process, spectral density, periodogram, data-taper, peak-insensitive spectral estimator.

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