The Annals of Statistics

Note on convergence rates of semiparametric estimators of dependence index

Peter Hall, Hira L. Koul, and Berwin A. Turlach

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Considerable recent attention has been devoted to semiparametric estimation of the dependence index, or the Hurst constant, using methods based on information in either frequency or time domains. Convergence rates of estimators in the frequency domain have been derived, and in the present paper we obtain them for estimators in the time domain. It is shown that the latter can have superior performance for moderate-range time series, but are inferior in the context of long-range dependence.

Article information

Ann. Statist., Volume 25, Number 4 (1997), 1725-1739.

First available in Project Euclid: 9 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62G05: Estimation

Autocovariance Hurst constant long-memory time series long-range dependence semiparametric inference stationary process Gaussian process short-range dependence


Hall, Peter; Koul, Hira L.; Turlach, Berwin A. Note on convergence rates of semiparametric estimators of dependence index. Ann. Statist. 25 (1997), no. 4, 1725--1739. doi:10.1214/aos/1031594739.

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  • BERAN, J. 1992. Statistical methods for data with long-range dependence with discussion. Statist. Sci. 7 404 427. Z.
  • BINGHAM, N. H., GOLDIE, C. M. and TEUGELS, J. L. 1987. Regular Variation. Cambridge Univ. Press. Z.
  • CHENG, B. and ROBINSON, P. M. 1994. Semiparametric estimation from time series with long-range dependence. J. Econometrics 64 335 353. Z.
  • DAHLHAUS, R. 1989. Efficient parameter estimation for self-similar processes. Ann. Statist. 17 1749 1766. Z.
  • DELGADO, M. A. and ROBINSON, P. M. 1994. New methods for the analysis of long-memory time series: application to Spanish inflation. J. Forecasting 13 97 107.
  • 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. Z.
  • GEWEKE, J. and PORTER-HUDAK, S. 1983. The estimation of long-memory time series models. J. Time Ser. Anal. 4 221 238. Z.
  • GIRAITIS, L. and SURGAILIS, D. 1990. A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asy mptotic normality of Whittle's estimate. Probab. Theory Related Fields 86 87 104. Z.
  • HALL, P., KOUL, H. L. and TURLACH, B. A. 1995. On estimating the self-similarity index for a long-range dependent process. Research Report No. SRR-32, Centre for Mathematics and Its Applications, Australian National Univ. Z.
  • ROBINSON, P. M. 1994a. Semiparametric analysis of long-memory time series. Ann. Statist. 22 515 539. Z.
  • ROBINSON, P. M. 1994b. Rates of convergence and optimal spectral bandwidth for long range dependence. Probab. Theory Related Fields 99 443 473. Z.
  • ROBINSON, P. M. 1994c. Time series with strong dependence. In Advances in Econometrics, Z. Sixth World Congress C. A. Sims, ed. 1 47 95. Cambridge Univ. Press. Z.
  • ROBINSON, P. M. 1995a. Log-periodogram regression of time series with long range dependence. Ann. Statist. 23 1048 1072. Z.
  • ROBINSON, P. M. 1995b. Gaussian semiparametric estimation of long range time dependence. Ann. Statist. 23 1630 1661. Z.
  • TAQQU, M. S. 1975. Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287 302. Z.
  • YAJIMA, Y. 1985. On estimation of long-memory time series models. Austral. J. Statist. 27 303 320. Z.
  • YAJIMA, Y. 1988. On estimation of a regression model with long-memory stationary errors. Ann. Statist. 16 791 807.