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A limit theory for long-range dependence and statistical inference on related models. (English) Zbl 0873.62096

Summary: This paper provides limit theorems for multivariate, possibly non-Gaussian stationary processes whose spectral density matrices may have singularities not restricted at the origin, applying those limiting results to the asymptotic theory of parameter estimation and testing for statistical models of long-range dependent processes. The central limit theorems are proved based on the assumption that the innovations of the stationary processes satisfy certain mixing conditions for their conditional moments, and the usual assumptions of exact martingale difference or the (transformed) Gaussianity for the innovation process are dispensed with.
For the proofs of convergence of the covariances of quadratic forms, the concept of the multiple Fejér kernel is introduced. For the derivation of the asymptotic properties of the quasi-likelihood estimate and the quasi-likelihood ratio, the bracketing function approach is used instead of conventional regularity conditions on the model spectral density.

MSC:

62M15 Inference from stochastic processes and spectral analysis
60F05 Central limit and other weak theorems
60G10 Stationary stochastic processes
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
Full Text: DOI

References:

[1] Cox, D. R. (1984). Long-range dependence: a review. In Statistics: An Appraisal (H. A. David and H. T. David, eds.) 55-73. Iowa State Univ. Press.
[2] Cox, D. R. (1991). Long-range dependence, non-linearity and time irreversibility. J. Time Ser. Anal. 12 329-335. · Zbl 0735.62088 · doi:10.1111/j.1467-9892.1991.tb00087.x
[3] Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17 1749-1766. · Zbl 0703.62091 · doi:10.1214/aos/1176347393
[4] Daniels, H. E. (1961). The asy mptotic efficiency of a maximum likelihood estimator. Proc. Fourth Berkeley Sy mp. Math. Statist. Probab. 1 151-163. Univ. California Press, Berkeley. · Zbl 0166.14802
[5] Dunsmuir, W. (1979). A central limit theorem for parameter estimation in stationary vector time series and its application to models for a signal observed with noise. Ann. Statist. 7 490-506. · Zbl 0406.62068 · doi:10.1214/aos/1176344671
[6] Dunsmuir, W. and Hannan, E. J. (1976). Vector linear time series models. Adv. in Appl. Probab. 8 339-364. JSTOR: · Zbl 0327.62055 · doi:10.2307/1425908
[7] Dzhaparidze, K. O. (1974). A new method for estimating spectral parameters of a stationary regular time series. Theory Probab. Appl. 19 122-132. · Zbl 0305.62061 · doi:10.1137/1119009
[8] Findley, D. F. and Wei, C. Z. (1993). Moment bound useful for deriving time series model selection procedures. Statist. Sinica 3 453-480. · Zbl 0822.62074
[9] Fox, R. and Taqqu, M. S. (1986). Large sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist. 14 517-532. · Zbl 0606.62096 · doi:10.1214/aos/1176349936
[10] Fox, R. and Taqqu, M. S. (1987). Central limit theorems for quadratic forms in random variables having long-range dependence. Probab. Theory Related Fields 74 213-240. · Zbl 0586.60019 · doi:10.1007/BF00569990
[11] Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asy mptotical normality of Whittle’s estimate. Probab. Theory Related Fields 86 87-104. · Zbl 0717.62015 · doi:10.1007/BF01207515
[12] Granger, C. W. J. and Joy eux, R. (1980). An introduction to long-memory time series and fractional differencing. J. Time Ser. Anal. 1 15-29. · Zbl 0503.62079 · doi:10.1111/j.1467-9892.1980.tb00297.x
[13] Hannan, E. J. (1970). Multiple Time Series. Wiley, New York. · Zbl 0211.49804
[14] Hannan, E. J. (1973). The asy mptotic theory of linear time-series models. J. Appl. Probab. 10 130-145. JSTOR: · Zbl 0261.62073 · doi:10.2307/3212501
[15] Hey de, C. C. and Gay, G. (1993). Smoothed periodogram asy mptotics and estimation for processes and fields with possible long-range dependence. Stochastic Process. Appl. 45 169-182. · Zbl 0771.60021 · doi:10.1016/0304-4149(93)90067-E
[16] Hosoy a, Y. (1974). Estimation problems on stationary time series models. Ph.D. dissertation, Dept. Statistics, Yale Univ. Hosoy a, Y. (1989a). The bracketing condition for limit theorems on stationary linear processes. Ann. Statist. 17 401-418. Hosoy a, Y. (1989b). Hierarchical statistical models and a generalized likelihood ratio test. J. Roy. Statist. Soc. Ser. B 51 435-447.
[17] Hosoy a, Y. (1993). Limit theorems for statistical inference on stationary processes with strong dependence. In Statistical Sciences and Data Analy sis. Proceedings of the Third Pacific Area Statistical Conference (K. Matsusita, M. L. Puri and T. Hay akawa, eds.) 151-163. Interscience, Utrecht. · Zbl 0854.62080
[18] Hosoy a, Y. and Taniguchi, M. (1982). A central limit theorem for stationary processes and the parameter estimation of linear processes. Ann. Statist. 10 132-153. · Zbl 0484.62102 · doi:10.1214/aos/1176345696
[19] Hosoy a, Y. and Taniguchi, M. (1993). Correction to Y. Hosoy a and M. Taniguchi. Ann. Statist. 21 115-117. · Zbl 0778.62085 · doi:10.1214/aos/1176349167
[20] Huber, P. (1967). The behavior of maximum-likelihood estimates under nonstandard conditions. Proc. Fifth Berkeley Sy mp. Math. Statist. Probab. 1 221-231. Univ. California Press, Berkeley. · Zbl 0212.21504
[21] Mandelbrot, B. B. and Wallis, J. R. (1969). Some long-run properties of geophysical records. Water Resource Research 5 321-340.
[22] Pollard, D. (1985). New way s to prove central limit theorems. Econometric Theory 1 295-314.
[23] Robinson, P. M. (1995). Gaussian semiparametric estimation of long range dependence. Ann. Statist. 23 1630-1661. · Zbl 0843.62092 · doi:10.1214/aos/1176324317
[24] Rozanov, Y. A. (1967). Stationary Random Process. Holden-Day, San Francisco. · Zbl 0152.16302
[25] Yajima, Y. (1985). On estimation of long-memory time series models. Austral. J. Statist. 27 303- 320. · Zbl 0584.62142 · doi:10.1111/j.1467-842X.1985.tb00576.x
[26] Whittle, P. (1952). Some results in time series analysis. Skand. Aktuarietidskr. 35 48-60. · Zbl 0049.37301
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