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Efficient estimation of seasonal long-range dependent processes. (English) Zbl 1097.62092

A class of Gaussian seasonal long-memory processes is considered with the spectral density \[ f(\omega)=H(\omega)| \omega| ^{-\alpha}\prod_{i=1}^r \prod_{j=1}^m| \omega-\omega_{ij}| ^{-\alpha_i}, \] where \(H\) is symmetric, strictly positive, continuous and bounded, and \(\omega_{ij}\) are known poles. \(f\) is assumed to be known up to a parameter \(\vartheta\) from a finite-dimensional compact set. Generalized ARMA and seasonal fractionally integrated ARMA models satisfy this specification. Consistency, asymptotic normality and Fisher’s efficiency of the maximum likelihood estimate for \(\vartheta\) are demonstrated. Results of simulations and application to Internet traffic data are presented.

MSC:

62M15 Inference from stochastic processes and spectral analysis
62F12 Asymptotic properties of parametric estimators
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F10 Point estimation
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[1] M. Abrahams, and A. Dempster(1979 ) Research on seasonal analysis, progress report on the ASA/Census Project on seasonal adjustment . Technical Report, Dept. Statistics, Harvard University.
[2] DOI: 10.1111/1467-9892.00170 · Zbl 0974.62079
[3] Basawa I. V., Sankhya, Series A 38 pp 259– (1976)
[4] Beran J., Statistics for Long Memory Processes (1994) · Zbl 0869.60045
[5] DOI: 10.1111/1467-9892.00275 · Zbl 1062.62164
[6] Brockwell P. J., Times Series: Theory and Methods, 2nd edn (1991) · Zbl 0709.62080
[7] DOI: 10.1214/aos/1028144856 · Zbl 0929.62091
[8] Chung C. F., Journal of Time Series Analysis 17 pp 111– (1996)
[9] Dahlhaus R., Annals Statistics 17 pp 1749– (1989)
[10] DOI: 10.1007/BF00569990 · Zbl 0586.60019
[11] DOI: 10.1214/aos/1013699989 · Zbl 1012.62098
[12] Giraitis L., Lithuanian Mathematical Journal 35 pp 53– (1995)
[13] Gray H. L., Journal of Time Series Analysis 10 pp 233– (1989)
[14] Hannan E.J., Journal of Applied Probability 10 pp 130– (1973)
[15] Hassler U., Journal of Time Series Analysis 15 pp 19– (1994) · Zbl 0794.62059
[16] Hassler U., Journal of Business Economics and Statistics 13 pp 37– (1995)
[17] A. J. Jonas(1979 ) Persistent Memory Random Processes. Unpublished Ph.D. dissertation , Dept. Statistics, Harvard University.
[18] DOI: 10.1111/1467-9892.00173 · Zbl 0974.62083
[19] Ling S., Journal of the American Statistical Association 92 pp 1184– (1997)
[20] DOI: 10.1029/2000WR900012
[21] M. Ooms(1995 ) Flexible seasonal long memory and economic time series . Technical Report, Econometric Institute, Erasmus University.
[22] DOI: 10.1023/A:1009967932430 · Zbl 0979.60011
[23] Ould Haye M., European Series in Applied and Industrial Mathematics (ESAIM): P&S 6 pp 293– (2002)
[24] Porter-Hudak S., Journal of the American Statistical Association 85 pp 338– (1990)
[25] DOI: 10.1016/0169-2070(93)90009-C
[26] Reisen V., Computational Statistics and Data Analysis (2005)
[27] Reisen V., Journal of Statistical Computation and Simulation (2005)
[28] Taniguchi M., Asymptotic Theory of Statistical Inference for Time Series (2000) · Zbl 0955.62088
[29] Velasco C., Journal of American Statistical Association 95 pp 1229– (2000)
[30] DOI: 10.1111/j.1467-9892.1998.00105.x · Zbl 1017.62083
[31] Yajima Y., Australian Journal of Statistics 27 pp 303– (1985)
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