zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A wavelet Whittle estimator of the memory parameter of a nonstationary Gaussian time series. (English) Zbl 1142.62062
Summary: We consider a time series $X=\{X_k$, $k\in \Bbb Z\}$ with memory parameter $d_{0}\in \Bbb R$. 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:
62M10Time series, auto-correlation, regression, etc. (statistics)
62G05Nonparametric estimation
42C40Wavelets and other special systems
62M15Spectral analysis of processes
62G20Nonparametric asymptotic efficiency
WorldCat.org
Full Text: DOI arXiv
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 · http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf
[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 · http://links.jstor.org/sici?sici=1061-8600%28199603%295%3A1%3C26%3AWAASOS%3E2.0.CO%3B2-Y&origin=euclid
[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.