Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-28T07:55:31.603Z Has data issue: false hasContentIssue false
  • English
  • Français

On a spectral density estimate obtained by averaging periodograms

Published online by Cambridge University Press:  14 July 2016

Rainer Dahlhaus
Rainer Dahlhaus*
Affiliation:
Universität Essen
*
∗Postal address: Universität Essen, Fachbereich Mathematik, Postfach 103764, 4300 Essen 1, W. Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

A spectral density statistic obtained by averaging periodograms over overlapping time intervals is considered where the periodograms are calculated using a data window. The asymptotic mean square error of this estimate for scale parameter windows is determined and, as an example, it is shown that the use of the Tukey–Hanning window leads partially to a smaller mean square error than a window suggested by Kolmogorov and Zhurbenko. Furthermore the Tukey–Hanning window is independent of the unknown spectral density, which is not the case for the Kolmogorov–Zhurbenko window. The mean square error of this estimate is also less than the mean square error of commonly used window estimates. Finally, a central limit theorem for the estimate is established.

Keywords

STATIONARY PROCESSSPECTRAL ESTIMATEDATA WINDOWMEAN SQUARE ERRORASYMPTOTIC NORMALITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 3 , September 1985 , pp. 598 - 610
Copyright
Copyright © Applied Probability Trust 1985 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bartlett, M. S. (1950) Periodogram analysis and continuous spectra. Biometrika 37, 1–16.Google Scholar
Bloomfield, P. (1976) Fourier Analysis of Time Series: An Introduction. Wiley Interscience, New York.Google Scholar
Brillinger, D. R. (1975) Time Series: Data Analysis and Theory. Holt Rinehart and Winston, New York.Google Scholar
Dahlhaus, R. (1983) Spectral analysis with tapered data. J. Time Series Anal. 4, 163–175.Google Scholar
Edwards, R. E. (1979) Fourier Series, volume I, 2nd edn. Springer-Verlag, New York.Google Scholar
Groves, W. G. and Hannan, E. J. (1968) Time series regression of sea level on weather. Rev. Geophysics 6, 129–174.Google Scholar
Kolmogorov, A. N. and Zhurbenko, I. G. (1978) Estimations of spectral functions of stochastic processes. In Proc. 11th European Meeting of Statisticians, Oslo, 36.Google Scholar
Nuttall, A. H. and Carter, G. C. (1982) Spectral estimation using combined time and lag weighting. Proc. IEEE 70, 1115–1125.CrossRefGoogle Scholar
Priestley, M. B. (1981) Spectral Analysis and Time Series, Vol. 1. Academic Press, London.Google Scholar
Thomson, D. J. (1977) Spectrum estimation techniques for characterization and development of WT4 waveguide, II. Bell System Tech. J. 56, 1983–2005.CrossRefGoogle Scholar
Welch, P. D. (1967) The use of the Fast Fourier Transform for estimation of spectra: A method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoustics AU-15, 70–74.Google Scholar
Zhurbenko, I. G. (1979a) On the efficiency of estimates of spectral density. Scand. J. Statist. 6, 49–56.Google Scholar
Zhurbenko, I. G. (1979b) On spectral density statistics obtained by means of a time shift. Sov. Math. Dokl. 20, 547–552.Google Scholar
Zhurbenko, I. G. (1980a) Optimal properties of some spectral-density statistics. Lith. Math. J. 20, 13–20.CrossRefGoogle Scholar
Zhurbenko, I. G. (1980b) On the efficiency of spectral density estimates of a stationary process. Theory Prob. Appl. 25, 466–480.Google Scholar
Zhurbenko, I. G. (1980c) A limit theorem for spectral density statistics with time shift. Ukrain. Math. J. 32, 316–326.Google Scholar
Zhurbenko, I. G. (1981) Statistics of the spectral densities of stationary stochastic processes. Sib. Math. J. 22, 680–700.Google Scholar
Zhurbenko, I. G. (1983) On the efficiency of spectral density estimates of a stationary process, II. Theory Prob. Appl. 28, 409–419.Google Scholar
Zhurbenko, I. G. and Kozhevnikova, I. A. (1982) Comparative characteristics of the statistics of spectral densities of stationary random processes. Probl. Inf. Transm. 18, 51–62.Google Scholar
Zhurbenko, I. G. and Zuev, N. M. (1975) On higher spectral densities of stationary processes with mixing. Ukrain. Math. J. 27, 364–373.CrossRefGoogle Scholar