The Annals of Statistics

A wavelet whittle estimator of the memory parameter of a nonstationary Gaussian time series

E. Moulines, F. Roueff, and M. S. Taqqu

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We consider a time series X={Xk, k∈ℤ} with memory parameter d0∈ℝ. This time series is either stationary or can be made stationary after differencing a finite number of times. We study the “local Whittle wavelet estimator” of the memory parameter d0. This is a wavelet-based semiparametric pseudo-likelihood maximum method estimator. The estimator may depend on a given finite range of scales or on a range which becomes infinite with the sample size. We show that the estimator is consistent and rate optimal if X is a linear process, and is asymptotically normal if X is Gaussian.

Article information

Ann. Statist., Volume 36, Number 4 (2008), 1925-1956.

First available in Project Euclid: 16 July 2008

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Zentralblatt MATH identifier

Primary: 62M15: Spectral analysis 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62G05: Estimation
Secondary: 62G20: Asymptotic properties 60G18: Self-similar processes

Long memory semiparametric estimation wavelet analysis


Moulines, E.; Roueff, F.; Taqqu, M. S. A wavelet whittle estimator of the memory parameter of a nonstationary Gaussian time series. Ann. Statist. 36 (2008), no. 4, 1925--1956. doi:10.1214/07-AOS527.

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  • [1] Abry, P. and Veitch, D. (1998). Wavelet analysis of long-range-dependent traffic. IEEE Trans. Inform. Theory 44 2–15.
  • [2] Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge Univ. Press.
  • [3] Cohen, A. (2003). Numerical Analysis of Wavelet Methods. North-Holland, Amsterdam.
  • [4] Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia.
  • [5] Faÿ, G., Roueff, F. and Soulier, P. (2007). Estimation of the memory parameter of the infinite-source Poisson process. Bernoulli. 13 473–491.
  • [6] Geweke, J. and Porter-Hudak, S. (1983). The estimation and application of long memory time series models. J. Time Ser. Anal. 4 221–238.
  • [7] Giraitis, L., Robinson, P. M. and Samarov, A. (1997). Rate optimal semiparametric estimation of the memory parameter of the Gaussian time series with long range dependence. J. Time Ser. Anal. 18 49–61.
  • [8] Hurvich, C. M., Moulines, E. and Soulier, P. (2002). The FEXP estimator for potentially nonstationary linear time series. Stoch. Proc. App. 97 307–340.
  • [9] Hurvich, C. M. and Ray, B. K. (1995). Estimation of the memory parameter for nonstationary or noninvertible fractionally integrated processes. J. Time Ser. Anal. 16 17–41.
  • [10] Kaplan, L. M. and Kuo, C.-C. J. (1993). Fractal estimation from noisy data via discrete fractional Gaussian noise (DFGN) and the Haar basis. IEEE Trans. Signal Process. 41 3554–3562.
  • [11] Künsch, H. R. (1987). Statistical aspects of self-similar processes. In Probability Theory and Applications. Proc. World Congr. Bernoulli Soc. 1 67–74. VNU Sci. Press, Utrecht.
  • [12] McCoy, E. J. and Walden, A. T. (1996). Wavelet analysis and synthesis of stationary long-memory processes. J. Comput. Graph. Statist. 5 26–56.
  • [13] Moulines, E., Roueff, F. and Taqqu, M. S. (2006). Central Limit Theorem for the log-regression wavelet estimation of the memory parameter in the Gaussian semi-parametric context. Fractals 15 301–313.
  • [14] Moulines, E., Roueff, F. and Taqqu, M. S. (2007). On the spectral density of the wavelet coefficients of long memory time series with application to the log-regression estimation of the memory parameter. J. Time Ser. Anal. 28.
  • [15] Robinson, P. M. (1995). Gaussian semiparametric estimation of long range dependence. Ann. Statist. 23 1630–1661.
  • [16] Robinson, P. M. (1995). Log-periodogram regression of time series with long range dependence. Ann. Statist. 23 1048–1072.
  • [17] Robinson, P. M. and Henry, M. (2003). Higher-order kernel semiparametric M-estimation of long memory. J. Econometrics 114 1–27.
  • [18] Rosenblatt, M. (1985). Stationary Sequences and Random Fields. Birkhäuser, Boston.
  • [19] Roughan, M., Veitch, D. and Abry, P. (2000). Real-time estimation of the parameters of long-range dependence. IEEE/ACM Transactions on Networking 8 467–478.
  • [20] Shimotsu, K. and Phillips, P. C. B. (2005). Exact local Whittle estimation of fractional integration. Ann. Statist. 33 1890–1933.
  • [21] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Univ. Press.
  • [22] Velasco, C. (1999). Gaussian semiparametric estimation of non-stationary time series. J. Time Ser. Anal. 20 87–127.
  • [23] Wornell, G. W. and Oppenheim, A. V. (1992). Estimation of fractal signals from noisy measurements using wavelets. IEEE Trans. Signal Process. 40 611–623.