• Bernoulli
  • Volume 6, Number 5 (2000), 845-869.

The multiple change-points problem for the spectral distribution

Marc Lavielle and Carenne Ludeña

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We consider the problem of detecting an unknown number of change-points in the spectrum of a second-order stationary random process. To reach this goal, some maximal inequalities for quadratic forms are first given under very weak assumptions. In a parametric framework, and when the number of changes is known, the change-point instants and the parameter vector are estimated using the Whittle pseudo-likelihood of the observations. A penalized minimum contrast estimate is proposed when the number of changes is unknown. The statistical properties of these estimates hold for strongly mixing and also long-range dependent processes. Estimation in a nonparametric framework is also considered, by using the spectral measure function. We conclude with an application to electroencephalogram analysis.

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Bernoulli, Volume 6, Number 5 (2000), 845-869.

First available in Project Euclid: 6 April 2004

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detection of change-points long range dependence maximal inequality nonparametric spectral estimation penalized minimum contrast estimate quadratic forms Whittle likelihood


Lavielle, Marc; Ludeña, Carenne. The multiple change-points problem for the spectral distribution. Bernoulli 6 (2000), no. 5, 845--869.

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  • [1] Akaike, H. (1974) A new look at the statistical model identification. IEEE Trans. Automat. Control, 19, 716-723.
  • [2] Basseville, M. and Nikiforov, N. (1993) The Detection of Abrupt Changes - Theory and Applications. Englewood Cliffs, NJ: Prentice Hall.
  • [3] Biscay, R., Lavielle, M., González, A., Clark, I. and Valdés, P. (1995) Maximum a posteriori estimation of change points in the EEG. Int. J. of Bio-Medical Computing, 38, 189-196.
  • [4] Brodsky, B. and Darkhovsky, B. (1993) Nonparametric Methods in Change-Point Problems. Dordrecht: Kluwer Academic Publishers.
  • [5] Dacunha-Castelle, D. and Gassiat, E. (1997) The estimation of the order of a mixture model. Bernoulli, 3, 279-299.
  • [6] Dahlhaus, R. (1997) Fitting time series models to nonstationary processes. Ann. Statist., 25, 1-37.
  • [7] Fox, R. and Taqqu, M. (1987) Central limit theorems for quadratic forms in random variables having long range dependence. Probab. Theory Related Fields, 74, 213-240.
  • [8] Giraitis, L. and Leipus, R. (1990) Functional CLT for nonparametric estimates of the spectrum and change-point problem for the spectral function. Lithuanian Math. J., 30, 674-679.
  • [9] Giraitis, L. and Leipus, R. (1992) Testing and estimating in the change-point problem of the spectral function. Lithuanian Math. J., 32, 20-38.
  • [10] Giraitis, L. and Surgailis, D. (1990) A central limit theorem for quadratic forms in strongly dependent linear vraiables and application to asymptotical normality of Whittle's estimate. Probab. Theory Related Fields, 86, 87-104.
  • [11] Hannan, J. (1980) The estimation of the order of an ARMA process. Ann. Statist., 8, 1071-1081.
  • [12] Lavielle, M. (1999) Detection of multiple change in a sequence of dependent variables. Stochastic Process. Appl., 83, 79-102.
  • [13] Lavielle, M. and Moulines, E. (1999) Least squares estimation of an unknown number of shifts in a time series. J. Time Series Anal. To appear.
  • [14] Ludena, C. and Lavielle, M. (1999) Some comments on the Whittle estimator for strongly dependent Gaussian fields. Scand. J. Statist., 26, 433-450.
  • [15] Móricz, F., Serfling, R. and Stout, W. (1982) Moment and probability bounds with quasi-superadditive structure for the maximum partial sum. Ann. Probab., 10, 1032-1040.
  • [16] Picard, D. (1985) Testing and estimating change points in time series. J. Appl. Probab., 17, 841-867.
  • [17] Taqqu, M., Beran, J., Sherman, R. and Willenger, W. (1995) Long range dependence in variable-bitrate video traffic. IEEE Trans. Commun., 43, 1566-1579.
  • [18] Terrin, N. and Taqqu, M. (1991) A non central limit theorem for quadratic forms of Gaussian stationary sequences. J. Theoret. Probab., 3, 449-475.