The Annals of Statistics

Autoregressive-aided periodogram bootstrap for timeseries

Jens-Peter Kreiss and Efstathios Paparoditis

Full-text: Open access

Abstract

A bootstrap methodology for the periodogram of a stationary process is proposed which is based on a combination of a time domain parametric and a frequency domain nonparametric bootstrap. The parametric fit is used to generate periodogram ordinates that imitate the essential features of the data and the weak dependence structure of the periodogram while a nonparametric (kernel-based) correction is applied in order to catch features not represented by the parametric fit. The asymptotic theory developed shows validity of the proposed bootstrap procedure for a large class of periodogram statistics. For important classes of stochastic processes, validity of the new procedure is also established for periodogram statistics not captured by existing frequency domain bootstrap methods based on independent periodogram replicates.

Article information

Source
Ann. Statist., Volume 31, Number 6 (2003), 1923-1955.

Dates
First available in Project Euclid: 16 January 2004

Permanent link to this document
https://projecteuclid.org/euclid.aos/1074290332

Digital Object Identifier
doi:10.1214/aos/1074290332

Mathematical Reviews number (MathSciNet)
MR2036395

Zentralblatt MATH identifier
1042.62081

Subjects
Primary: 62G09: Resampling methods
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Keywords
Bootstrap periodogram nonparametric estimators ratio statistics spectral means

Citation

Kreiss, Jens-Peter; Paparoditis, Efstathios. Autoregressive-aided periodogram bootstrap for timeseries. Ann. Statist. 31 (2003), no. 6, 1923--1955. doi:10.1214/aos/1074290332. https://projecteuclid.org/euclid.aos/1074290332


Export citation

References

  • Anderson, T. W. (1971). The Statistical Analysis of Time Series. Wiley, New York.
  • Bauer, H. (1974). Wahrscheinlichkeitstheorie und Grundzüge der Masstheorie. de Gruyter, Berlin.
  • Baxter, G. (1962). An asymptotic result for the finite predictor. Math. Scand. 10 137--144.
  • Beltrão, K. I. and Bloomfield, P. (1987). Determining the bandwidth of a kernel spectrum estimate. J. Time Ser. Anal. 8 21--38.
  • Berk, K. N. (1974). Consistent autoregressive spectral estimates. Ann. Statist. 2 489--502.
  • Bickel, P. J. and Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9 1196--1217.
  • Brillinger, D. (1981). Time Series: Data Analysis and Theory. Holden-Day, San Francisco.
  • Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
  • Dahlhaus, R. (1985). Asymptotic normality of spectral estimates. J. Multivariate Anal. 16 412--431.
  • Dahlhaus, R. and Janas, D. (1996). A frequency domain bootstrap for ratio statistics in time series analysis. Ann. Statist. 24 1934--1963.
  • Franke, J. and Härdle, W. (1992). On bootstrapping kernel spectral estimates. Ann. Statist. 20 121--145.
  • Hannan, E. J. and Kavalieris, L. (1986). Regression, autoregression models. J. Time Ser. Anal. 7 27--49.
  • Hurvich, C. M. and Zeger, S. L. (1987). Frequency domain bootstrap methods for time series. Technical Report 87--115, Graduate School of Business Administration, New York Univ.
  • Janas, D. and Dahlhaus, R. (1994). A frequency domain bootstrap for time series. In Computationally Intensive Statistical Methods. Proc. 26th Symposium on the Interface (J. Sall and A. Lehman, eds.) 423--425. Interface Foundation of North America, Fairfax Station, VA.
  • Kreiss, J.-P. (1999). Residual and wild bootstrap for infinite order autoregression. Unpublished manuscript.
  • Kreiss, J.-P. and Franke, J. (1992). Bootstrapping stationary autoregressive moving-average models. J. Time Ser. Anal. 13 297--317.
  • Nordgaard, A. (1992). Resampling stochastic process es using a bootstrap approach. In Lecture Notes in Econom. and Math. Systems 376 181--185.
  • Paparoditis, E. (2002). Frequency domain bootstrap for time series. In Empirical Process Techniques for Dependent Data (H. Dehling, T. Mikosch and M. Sørensen, eds.) 365--381. Birkhäuser, Boston.
  • Paparoditis, E. and Politis, D. N. (1999). The local bootstrap for the periodogram. J. Time Ser. Anal. 20 193--222.
  • Press, H. and Tukey, J. W. (1956). Power spectral methods of analysis and their application to problems in airplane dynamics. Bell System Monographs 2606.
  • Priestley, M. B. (1981). Spectral Analysis and Time Series. Academic Press, New York.
  • Robinson, P. (1991). Automatic frequency domain inference on semiparametric and nonparametric models. Econometrica 59 1329--1363.
  • Romano, J. P. (1988). Bootstrapping the mode. Ann. Inst. Statist. Math. 40 565--586.
  • Rosenblatt, M. (1985). Stationary Sequences and Random Fields. Birkhäuser, Boston.
  • Shibata, R. (1981). An optimal autoregressive spectral estimate. Ann. Statist. 9 300--306.
  • Shiryayev, A. N. (1984). Probability. Springer, New York.
  • Theiler, J., Paul, L. S. and Rubin, D. M. (1994). Detecting nonlinearity in data with long coherence times. In Time Series Prediction (A. Weigend and N. Gershenfeld, eds.). Addison--Wesley, Reading, MA.