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Consistency For Non‐Linear Functions Of The Periodogram Of Tapered Data

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  • Daniel Janas
  • Rainer von Sachs

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.

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  • Daniel Janas & Rainer von Sachs, 1995. "Consistency For Non‐Linear Functions Of The Periodogram Of Tapered Data," Journal of Time Series Analysis, Wiley Blackwell, vol. 16(6), pages 585-606, November.
    • Handle: RePEc:bla:jtsera:v:16:y:1995:i:6:p:585-606
    DOI: 10.1111/j.1467-9892.1995.tb00257.x
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    References listed on IDEAS

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    1. Daniel Janas, 1994. "Edgeworth expansions for spectral mean estimates with applications to Whittle estimates," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(4), pages 667-682, December.
    2. Chen Zhao‐Guo & E. J. Hannan, 1980. "The Distribution Of Periodogram Ordinates," Journal of Time Series Analysis, Wiley Blackwell, vol. 1(1), pages 73-82, January.
    3. Dahlhaus, Rainer, 1985. "Asymptotic normality of spectral estimates," Journal of Multivariate Analysis, Elsevier, vol. 16(3), pages 412-431, June.
    4. Rainer Dahlhaus, 1983. "Spectral Analysis With Tapered Data," Journal of Time Series Analysis, Wiley Blackwell, vol. 4(3), pages 163-175, May.
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