A wavelet Whittle estimator of the memory parameter of a nonstationary Gaussian time series. (English) Zbl 1142.62062

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

Ann. Stat. 36, No. 4, 1925-1956 (2008).

Summary: We consider a time series \(X=\{X_k\), \(k\in \mathbb Z\}\) with memory parameter \(d_{0}\in \mathbb R\). 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 \(d_{0}\). 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.

MSC:

62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G05	Nonparametric estimation
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
62M15	Inference from stochastic processes and spectral analysis
62G20	Asymptotic properties of nonparametric inference

Keywords:

long memory; semiparametric estimation; wavelet analysis

References:

[1]	Abry, P. and Veitch, D. (1998). Wavelet analysis of long-range-dependent traffic. IEEE Trans. Inform. Theory 44 2-15. · Zbl 0905.94006 · doi:10.1109/18.650984
[2]	Boyd, S. and Vandenberghe, L. (2004). Convex Optimization . Cambridge Univ. Press. · Zbl 1058.90049
[3]	Cohen, A. (2003). Numerical Analysis of Wavelet Methods . North-Holland, Amsterdam. · Zbl 1038.65151
[4]	Daubechies, I. (1992). Ten Lectures on Wavelets . SIAM, Philadelphia. · Zbl 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. · Zbl 1127.62070 · doi:10.3150/07-BEJ5123
[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. · Zbl 0534.62062 · 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. · Zbl 0870.62073 · 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. · Zbl 1057.62074 · 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. · Zbl 0813.62081 · 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. · Zbl 0841.94012 · doi:10.1109/78.258096
[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. · Zbl 0673.62073
[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. JSTOR: · 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. · Zbl 1141.62073 · 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 . · Zbl 1150.62058 · 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. · Zbl 0843.62092 · doi:10.1214/aos/1176324317
[16]	Robinson, P. M. (1995). Log-periodogram regression of time series with long range dependence. Ann. Statist. 23 1048-1072. · Zbl 0838.62085 · doi:10.1214/aos/1176324636
[17]	Robinson, P. M. and Henry, M. (2003). Higher-order kernel semiparametric M -estimation of long memory. J. Econometrics 114 1-27. · Zbl 1023.62106 · doi:10.1016/S0304-4076(02)00208-7
[18]	Rosenblatt, M. (1985). Stationary Sequences and Random Fields . Birkhäuser, Boston. · Zbl 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. · Zbl 1081.62069 · doi:10.1214/009053605000000309
[21]	van der Vaart, A. W. (1998). Asymptotic Statistics . Cambridge Univ. Press. · Zbl 0910.62001 · doi:10.1017/CBO9780511802256
[22]	Velasco, C. (1999). Gaussian semiparametric estimation of non-stationary time series. J. Time Ser. Anal. 20 87-127. · Zbl 0922.62093 · 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.