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Gaussian estimation of parametric spectral density with unknown pole. (English) Zbl 1012.62098

Summary: We consider a parametric spectral density with power-law behavior about a fractional pole at the unknown frequency \(\omega\). The case of known \(\omega\), especially \(\omega =0\), is standard in the long memory literature. When \(\omega\) is unknown, asymptotic distribution theory for estimates of parameters, including the (long) memory parameter, is significantly harder. We study a form of Gaussian estimate. We establish \(n\)-consistency of the estimate of \(\omega\), and discuss its (non-standard) limiting distributional behavior. For the remaining parameter estimates, we establish \(\sqrt{n}\)-consistency and asymptotic normality.

MSC:

62M15 Inference from stochastic processes and spectral analysis
62F12 Asymptotic properties of parametric estimators
62E20 Asymptotic distribution theory in statistics
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[1] Andel, J. (1986). Long-memory time series models. Kybernetika 22 105-123. · Zbl 0607.62111
[2] Bloomfield, P. (1973). An exponential model for the spectrum of a scalar time series. Biometrika 60 217-226. JSTOR: · Zbl 0261.62074
[3] Box, G. E. P. and Jenkins, G. M. (1971). Time Series Analysis. Forecasting and Control. HoldenDay, San Francisco. Chung, C. F. (1996a). Estimating a generalized long memory process. J. Econometrics 73 237-259. Chung, C. F. (1996b). A generalized fractionally integrated autoregressive moving average process. J. Time Ser. Anal. 17 111-140.
[4] Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17 1749-1766. · Zbl 0703.62091
[5] 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
[6] Giraitis, L. and Leipus, R. (1995). A generalized fractionally differencing approach in longmemory modelling. Lithuanian Math. J. 35 65-81. · Zbl 0837.62066
[7] Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent random variables and its application to asymptotical normality of Whittle’s estimate. Probab. Theory Related Fields 86 87-104. · Zbl 0717.62015
[8] Gray, H. L., Zhang, N. and Woodward, W. A. (1989). On generalized fractional processes. J. Time Ser. Anal. 10 233-257. · Zbl 0685.62075
[9] Hannan, E. J. (1970). Multiple Time Series. Wiley, New York. Hannan, E. J. (1973a). The estimation of frequency. J. Appl. Probab. 10 513-519. Hannan, E. J. (1973b). The asymptotic theory of linear time series models. J. Appl. Probab. 10 130-145. · Zbl 0211.49804
[10] Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68 165-176. JSTOR: · Zbl 0464.62088
[11] Hosoya, Y (1997). Limit theory with long-range dependence and statistical inference of related models. Ann. Statist. 25 105-137. · Zbl 0873.62096
[12] Hidalgo, J. (1999). Semiparametric estimation of the location of the pole. Unpublished manuscript.
[13] Robinson, P. M. (1978). Alternative models for stationary stochastic processes. Stochastic Process. Appl. 8 141-152. · Zbl 0391.62069
[14] Robinson, P. M. (1994). Efficient tests of nonstationary hypotheses. J. Amer. Statist. Assoc. 89 1420-1437. Robinson, P. M. (1995a). Log-periodogram regression of time series with long range dependence. Ann. Statist. 23 1048-1072. Robinson, P. M. (1995b). Gaussian semiparametric estimation of long range dependence. Ann. Statist. 23 1630-1661. JSTOR: · Zbl 0813.62016
[15] Stout, W. F. (1974). Almost Sure Convergence. Academic Press, New York. · Zbl 0321.60022
[16] Yajima, Y. (1996). Estimation of the frequency of unbounded spectral densities. In Proceedings of the Business and Economic Statistical Section 4-7. Amer. Statist. Assoc., Alexandria, VA.
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