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(ω) be the spectral density of a Gaussian stationary process. Consider periodogram-based estimators of integrals of certain non-linear functions ζ of f(ω), like H_{T} := ∫_{−π}^{π} Λ(ω)ζ(I_{T} (ω))dω, where Λ(ω) is a bounded function of bounded variation, possibly depending on the sample size T. Then it is known that, under mild conditions on ζ, a central limit theorem holds for these statistics H_{T} if the non-tapered periodogram I_{T} (ω) is used. In particular, Taniguchi (1980, J. Appl. Probab., 17, 73–83) gave a consistent and asymptotic normal estimator of ∫_{−π}^{π} Λ(ω)Φ(f(ω))dω, choosing ζ to be a suitable transform of a given function Φ. In this work we shall generalize this result to statistics H_{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.

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