The Annals of Statistics

Gaussian estimation of parametric spectral density with unknown pole

L. Giraitis, J. Hidalgo, and P. M. Robinson

Full-text: Open access


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.

Article information

Ann. Statist., Volume 29, Number 4 (2001), 987-1023.

First available in Project Euclid: 14 February 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 60G18: Self-similar processes

Long range dependence unknown pole


Giraitis, L.; Hidalgo, J.; Robinson, P. M. Gaussian estimation of parametric spectral density with unknown pole. Ann. Statist. 29 (2001), no. 4, 987--1023. doi:10.1214/aos/1013699989.

Export citation


  • Andel, J. (1986). Long-memory time series models. Kybernetika 22 105-123.
  • Bloomfield, P. (1973). An exponential model for the spectrum of a scalar time series. Biometrika 60 217-226.
  • 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.
  • Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17 1749-1766.
  • 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.
  • Giraitis, L. and Leipus, R. (1995). A generalized fractionally differencing approach in longmemory modelling. Lithuanian Math. J. 35 65-81.
  • 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.
  • Gray, H. L., Zhang, N. and Woodward, W. A. (1989). On generalized fractional processes. J. Time Ser. Anal. 10 233-257.
  • 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.
  • Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68 165-176.
  • Hosoya, Y (1997). Limit theory with long-range dependence and statistical inference of related models. Ann. Statist. 25 105-137.
  • Hidalgo, J. (1999). Semiparametric estimation of the location of the pole. Unpublished manuscript.
  • Robinson, P. M. (1978). Alternative models for stationary stochastic processes. Stochastic Process. Appl. 8 141-152.
  • 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.
  • Stout, W. F. (1974). Almost Sure Convergence. Academic Press, New York.
  • 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.