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ESTIMATING NON-LINEAR FUNCTIONS OF THE SPECTRAL

DENSITY, USING A DATA-TAPER

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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 *H*_{T} := \int ^{pi}_{-pi} *Lambda*(*omega*)
*zeta*(*I*_{T}(*omega*))*d**omega*, 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 *H*_{T} if the non-tapered periodogram *I*_{T}(*omega*)
is used. In particular, Taniguchi (1980, *J. Appl.
Probab.*, **17**, 73-83) gave a consistent and asymptotic
normal estimator of \int ^{pi}_{-pi}*Lambda*(*omega*)
*Phi*(*f*(*omega*))d*omega*, choosing *zeta* to be a suitable
transform of a given function *Phi*. 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.

**Source**
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