Summary
For a realization of lengthn from a covariance stationary discrete time process with spectral density which behaves like λ1−2H as λ→0+ for 1/2<H<1 (apart from a slowly varying factor which may be of unknown form), we consider a discrete average of the periodogram across the frequencies 2πj/n,j=1,..., m, wherem→∞ andm/n→0 asn→∞. We study the rate of convergence of an analogue of the mean squared error of smooth spectral density estimates, and deduce an optimal choice ofm.
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Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation. New York: Cambridge University Press 1987
Brillinger, D.R.: Time series, data analysis and theory. New York: Holt, Rinehart and Winston 1975
Dahlhaus, R.: Efficient parameter estimation for self-similar processes. Ann. Stat.17, 1749–1766 (1989)
Dahlhaus, R.: Nonparametric high resolution spectral estimation. Probab. Theory Relat. Fields85, 147–180 (1990)
Fox, R., Taqqu, M.S.: Large-sample properties of parameter estimates for strongly dependent stationary Gaussian series. Ann. Stat.14, 517–532 (1986)
Giraitis, L., Surgailis, D.: A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittle's estimate. Probab. Theory Relat. Fields86, 87–104 (1990)
Grenander, U., Rosenblatt, M.: Statistical analysis of stationary time series. New York: Wiley 1957
Hall, P., Hart, J.: Convergence rates in density estimation for data from infinite-order moving average processes. Probab. Theory Relat. Fields87, 253–274 (1990)
Hannan, E.J.: Multiple time series. New York: Wiley 1970
Hannan, E.J.: The asymptotic theory of linear time series models. J. Appl. Probab.10, 130–145 (1973)
Robinson, P.M.: Nonparametric function estimation for long memory time series. In: Barnett, W. et al. (eds.) Nonparametric and semiparametric methods in econometrics and statistics (pp. 437–457). New York: Cambridge University Press 1991
Robinson, P.M.: Semiparametric analysis of long-memory time series. Ann. Stat. (forthcoming 1991)
Rosenblatt, M.: Limit theorems for Fourier transforms of functionals of Gaussian sequences. Z. Wahrscheinlichkeitstheor. Verw. Geb.55, 123–132 (1981)
Taqqu, M.S.: Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitstheor. Verw. Geb.31, 287–302 (1975)
Yong, C.H.: Asymptotic behaviour of trigonometric series. Hong Kong: Chinese University of Hong Kong 1974
Zurbenko, I.G.: The spectral analysis of time series. Amsterdam: North Holland 1986
Zygmund, A.: Trigonometric series. New York: Cambridge University Press 1986
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Robinson, P.M. Rates of convergence and optimal spectral bandwidth for long range dependence. Probab. Th. Rel. Fields 99, 443–473 (1994). https://doi.org/10.1007/BF01199901
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DOI: https://doi.org/10.1007/BF01199901