The Annals of Statistics

Estimation for almost periodic processes

Keh-Shin Lii and Murray Rosenblatt

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Abstract

Processes with almost periodic covariance functions have spectral mass on lines parallel to the diagonal in the two-dimensional spectral plane. Methods have been given for estimation of spectral mass on the lines of spectral concentration if the locations of the lines are known. Here methods for estimating the intercepts of the lines of spectral concentration in the Gaussian case are given under appropriate conditions. The methods determine rates of convergence sufficiently fast as the sample size n→∞ so that the spectral estimation on the estimated lines can then proceed effectively. This task involves bounding the maximum of an interesting class of non-Gaussian possibly nonstationary processes.

Article information

Source
Ann. Statist., Volume 34, Number 3 (2006), 1115-1139.

Dates
First available in Project Euclid: 10 July 2006

Permanent link to this document
https://projecteuclid.org/euclid.aos/1152540744

Digital Object Identifier
doi:10.1214/009053606000000218

Mathematical Reviews number (MathSciNet)
MR2278353

Zentralblatt MATH identifier
1113.62111

Subjects
Primary: 62M15: Spectral analysis 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62G05: Estimation 62M99: None of the above, but in this section

Keywords
Almost periodic covariance spectral estimation periodogram process maximum Gaussian and non-Gaussian process frequency detection and estimation

Citation

Lii, Keh-Shin; Rosenblatt, Murray. Estimation for almost periodic processes. Ann. Statist. 34 (2006), no. 3, 1115--1139. doi:10.1214/009053606000000218. https://projecteuclid.org/euclid.aos/1152540744


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References

  • Alekseev, V. G. (1988). Estimating the spectral densities of a Gaussian periodically correlated stochastic process. Problems Inform. Transmission 24 109--115.
  • Brillinger, D. (1975). Time Series: Data Analysis and Theory. Holt, Rinehart and Winston, New York.
  • Dandawate, A. V. and Giannakis, G. B. (1994). Nonparametric polyspectral estimators for $k$th order (almost) cyclostationary processes. IEEE Trans. Inform. Theory 40 67--84.
  • Dehay, D. and Leskow, J. (1996). Functional limit theory for the spectral covariance estimator. J. Appl. Probab. 33 1077--1092.
  • Gardner, W. A. (1991). Exploitation of spectral redundancy in cyclostationary signals. IEEE Signal Processing Magazine 8 (2) 14--36.
  • Gardner, W. A., ed. (1994). Cyclostationarity in Communications and Signal Processing. IEEE Press, New York.
  • Gerr, N. and Allen, J. (1994). The generalized spectrum and spectral coherence of a harmonizable time series. Digital Signal Processing 4 222--238.
  • Gerr, N. and Allen, J. (1994). Time-delay estimation for harmonizable signals. Digital Signal Processing 4 49--62.
  • Gladyshev, E. G. (1963). Periodically and almost-periodically correlated random processes with a continuous time parameter. Theory Probab. Appl. 8 173--177.
  • Hurd, H. (1989). Nonparametric time series analysis for periodically correlated processes. IEEE Trans. Inform. Theory 35 350--359.
  • Hurd, H. and Gerr, N. (1991). Graphical methods for determining the presence of periodic correlation. J. Time Ser. Anal. 12 337--350.
  • Hurd, H. and Leskow, J. (1992). Strongly consistent and asymptotically normal estimation of the covariance for almost periodically correlated processes. Statist. Decisions 10 201--225.
  • Hurd, H., Makagon, A. and Miamee, A. G. (2002). On AR(1) models with periodic and almost periodic coefficients. Stochastic Process. Appl. 100 167--185.
  • Leskow, J. and Weron, A. (1992). Ergodic behavior and estimation for periodically correlated processes. Statist. Probab. Lett. 15 299--304.
  • Lii, K.-S. and Rosenblatt, M. (2002). Spectral analysis for harmonizable processes. Ann. Statist. 30 258--297.
  • Loève, M. (1963). Probability Theory, 3rd ed. Van Nostrand, Princeton, NJ.
  • Lund, R., Hurd, H., Bloomfield, P. and Smith, R. (1995). Climatological time series with periodic correlation. J. Climate 8 2787--2809.
  • Tian, C. J. (1988). A limiting property of sample autocovariances of periodically correlated processes with application to period determination. J. Time Ser. Anal. 9 411--417.
  • Woodroofe, M. B. and Van Ness, J. W. (1967). The maximum deviation of sample spectral densities. Ann. Math. Statist. 38 1558--1569.
  • Yaglom, A. M. (1987). Correlation Theory of Stationary and Related Random Functions 1, 2. Springer, Berlin.,
  • Zygmund, A. (1959). Trigonometric Series 1, 2, 2nd ed. Cambridge Univ. Press.