CONSISTENCY FOR NON‐LINEAR FUNCTIONS OF THE PERIODOGRAM OF TAPERED DATA

Abstract

Abstract. In this paper we investigate the merits of using a data taper in non‐linear functional of the periodogram of a stationary time series. To this end, we show consistency for a general class of statistics of the form , where A(ω) is a function of bounded variation and where φ is allowed to be a non‐linear function of the periodogram IT(ω) of the tapered data. The key step in deriving our asymptotic results is an Edgeworth expansion for the finite Fourier transform of the tapered data, which do not have to follow a particular distribution (i.e. we allow for non‐Gaussianity). Important applications are the estimation of , choosing φ to be a suitable transform of a given function g (see Taniguchi, On estimation of the integrals of certain functions of spectral density. J. Appl. Prob. 17 (1980). 73–83), the peak‐insensitive spectrum estimator of von Sachs (Peak‐insensitive nonparametric spectrum estimation. J. Time Ser. Anal. 15 (1994), 429–52), where φ is chosen to be a bounded (robustifying) σ function, and the parametric approach of Chiu (Peak‐insensitive parametric spectrum estimation. Stoch. Proc. Appl. 35 (1990). 121–40) on robust estimation of the parameters of the continuous spectrum of the time series.