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



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

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

Abstract

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.

Primary Subjects: 62M15, 62M10, 62G05
Secondary Subjects: 62G20, 60G18
Keywords: Long memory; semiparametric estimation; wavelet analysis

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Alternatively, the document is available for a cost of $15. Select the "buy article" button below to purchase this document from a secured VeriSign, Inc. site.
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aos/1216237304
Digital Object Identifier: doi:10.1214/07-AOS527
Mathematical Reviews number (MathSciNet): MR2435460

References

[1] Abry, P. and Veitch, D. (1998). Wavelet analysis of long-range-dependent traffic. IEEE Trans. Inform. Theory 44 2–15.
Mathematical Reviews (MathSciNet): MR1486645
Digital Object Identifier: doi:10.1109/18.650984
[2] Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR2061575
Zentralblatt MATH: 1058.90049
[3] Cohen, A. (2003). Numerical Analysis of Wavelet Methods. North-Holland, Amsterdam.
Mathematical Reviews (MathSciNet): MR1990555
Zentralblatt MATH: 1038.65151
[4] Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia.
Mathematical Reviews (MathSciNet): MR1162107
Zentralblatt MATH: 0776.42018
[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.
Mathematical Reviews (MathSciNet): MR738585
Digital Object Identifier: doi:10.1111/j.1467-9892.1983.tb00371.x
[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.
Mathematical Reviews (MathSciNet): MR1437741
Digital Object Identifier: doi:10.1111/1467-9892.00038
[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.
Mathematical Reviews (MathSciNet): MR1875337
Digital Object Identifier: doi:10.1016/S0304-4149(01)00136-3
[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.
Mathematical Reviews (MathSciNet): MR1323616
Digital Object Identifier: doi:10.1111/j.1467-9892.1995.tb00221.x
[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.
Mathematical Reviews (MathSciNet): MR1380851
Digital Object Identifier: doi:10.2307/1390751
[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.
Mathematical Reviews (MathSciNet): MR2396718
Digital Object Identifier: doi:10.1142/S0218348X07003721
[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.
Mathematical Reviews (MathSciNet): MR2345656
Digital Object Identifier: doi:10.1111/j.1467-9892.2006.00502.x
[15] Robinson, P. M. (1995). Gaussian semiparametric estimation of long range dependence. Ann. Statist. 23 1630–1661.
Mathematical Reviews (MathSciNet): MR1370301
Digital Object Identifier: doi:10.1214/aos/1176324317
Project Euclid: euclid.aos/1176324317
[16] Robinson, P. M. (1995). Log-periodogram regression of time series with long range dependence. Ann. Statist. 23 1048–1072.
Mathematical Reviews (MathSciNet): MR1345214
Digital Object Identifier: doi:10.1214/aos/1176324636
Project Euclid: euclid.aos/1176324636
[17] Robinson, P. M. and Henry, M. (2003). Higher-order kernel semiparametric M-estimation of long memory. J. Econometrics 114 1–27.
Mathematical Reviews (MathSciNet): MR1962371
Digital Object Identifier: doi:10.1016/S0304-4076(02)00208-7
[18] Rosenblatt, M. (1985). Stationary Sequences and Random Fields. Birkhäuser, Boston.
Mathematical Reviews (MathSciNet): MR885090
Zentralblatt MATH: 0597.62095
[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.
Mathematical Reviews (MathSciNet): MR2166565
Digital Object Identifier: doi:10.1214/009053605000000309
Project Euclid: euclid.aos/1123250232
[21] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR1652247
Zentralblatt MATH: 0910.62001
[22] Velasco, C. (1999). Gaussian semiparametric estimation of non-stationary time series. J. Time Ser. Anal. 20 87–127.
Mathematical Reviews (MathSciNet): MR1678573
Digital Object Identifier: doi:10.1111/1467-9892.00127
[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.

2008 © Institute of Mathematical Statistics