The Annals of Statistics

Valid asymptotic expansions for the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process

Offer Lieberman, Judith Rousseau, and David M. Zucker

Full-text: Open access


We establish the validity of an Edgeworth expansion to the distribution of the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process. The result covers ARFIMA-type models, including fractional Gaussian noise. The method of proof consists of three main ingredients: (i) verification of a suitably modified version of Durbin's general conditions for the validity of the Edgeworth expansion to the joint density of the log-likelihood derivatives; (ii) appeal to a simple result of Skovgaard to obtain from this an Edgeworth expansion for the joint distribution of the log-likelihood derivatives; (iii) appeal to and extension of arguments of Bhattacharya and Ghosh to accomplish the passage from the result on the log-likelihood derivatives to the result for the maximum likelihood estimators. We develop and make extensive use of a uniform version of a theorem of Dahlhaus on products of Toeplitz matrices; the extension of Dahlhaus' result is of interest in its own right. A small numerical study of the efficacy of the Edgeworth expansion is presented for the case of fractional Gaussian noise.

Article information

Ann. Statist., Volume 31, Number 2 (2003), 586-612.

First available in Project Euclid: 22 April 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E17: Approximations to distributions (nonasymptotic) 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Edgeworth expansions long memory processes ARFIMA models


Lieberman, Offer; Rousseau, Judith; Zucker, David M. Valid asymptotic expansions for the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process. Ann. Statist. 31 (2003), no. 2, 586--612. doi:10.1214/aos/1051027882.

Export citation


  • ANDERSON, T. W. (1984). An Introduction to Multivariate Statistical Analy sis, 2nd ed. Wiley, New York.
  • BARNDORFF-NIELSEN, O. E. and COX, D. R. (1989). Asy mptotic Techniques for Use in Statistics. Chapman and Hall, London.
  • BARNDORFF-NIELSEN, O. E. and COX, D. R. (1994). Inference and Asy mptotics. Chapman and Hall, London.
  • BARNDORFF-NIELSEN, O. E. and HALL, P. (1988). On the level-error after Bartlett adjustment of the likelihood ratio statistic. Biometrika 75 374-378.
  • BHATTACHARy A, R. N. and GHOSH, J. K. (1978). On the validity of the formal Edgeworth expansion. Ann. Statist. 6 434-451.
  • BHATTACHARy A, R. N. and RANGA RAO, R. (1976). Normal Approximation and Asy mptotic Expansions. Wiley, New York.
  • BOX, G. E. P. and JENKINS, G. M. (1976). Time Series Analy sis: Forecasting and Control, rev. ed. Holden-Day, San Francisco.
  • BROCKWELL, P. J. and DAVIS, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
  • CHAMBERS, J. M. (1967). On methods of asy mptotic approximation for multivariate distributions. Biometrika 54 367-383.
  • CHIBISOV, D. M. (1973). An asy mptotic expansion for a class of estimators containing maximum likelihood estimators. Theory Probab. Appl. 18 295-303.
  • DAHLHAUS, R. (1988). Small sample effects in time series analysis: A new asy mptotic theory and a new estimate. Ann. Statist. 16 808-841.
  • DAHLHAUS, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17 1749-1766.
  • DURBIN, J. (1980). Approximations for densities of sufficient estimators. Biometrika 67 311-333.
  • 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.
  • 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.
  • GÖTZE, F. and HIPP, C. (1994). Asy mptotic distribution of statistics in time series. Ann. Statist. 22 2062-2088.
  • 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.
  • GRENANDER, U. and SZEGÖ, G. (1958). Toeplitz Forms and Their Applications. Univ. California Press, Berkeley. (2nd ed., Chelsea, New York, 1984.)
  • HORN, R. A. and JOHNSON, C. R. (1985). Matrix Analy sis. Cambridge Univ. Press.
  • HOSKING, J. R. M. (1981). Fractional differencing. Biometrika 68 165-176.
  • HURST, H. E. (1951). Long-term storage capacity of reservoirs. Trans. Amer. Soc. Civil Engin. 116 770-808.
  • JAMES, G. S. (1955). Cumulants of a transformed variate. Biometrika 42 529-531.
  • JAMES, G. S. (1958). On moments and cumulants of sy stems of statistics. Sanky¯a 20 1-30.
  • JAMES, G. S. and MAy NE, A. J. (1962). Cumulants of functions of a random variable. Sanky¯a Ser. A 24 47-54.
  • JOHNSON N. L, KOTZ, S. and BALAKRISHNAN, N. (1997). Continuous Univariate Distributions 2, 2nd ed. Wiley, New York.
  • LAHIRI, S. N. (1993). Refinements in asy mptotic expansions for sums of weakly dependent random vectors. Ann. Probab. 21 791-799.
  • LIEBERMAN, O., ROUSSEAU, J. and ZUCKER, D. M. (2000). Small-sample likelihood-based inference in the ARFIMA model. Econometric Theory 16 231-248.
  • MANDELBROT, B. B. and VAN NESS, J. W. (1968). Fractional Brownian motions, fractional noise and applications. SIAM Rev. 10 422-437.
  • PEERS, H. W. and IQBAL, M. (1985). Asy mptotic expansions for confidence limits in the presence of nuisance parameters, with applications. J. Roy. Statist. Soc. Ser. B 47 547-554.
  • ROBINSON, P. M. (1995). Time series with strong dependence. In Advances in Econometrics. Sixth World Congress (C. A. Sims, ed.) 1 47-95. Cambridge Univ. Press.
  • ROCKE, D. M. (1989). Bootstrap Bartlett adjustment in seemingly unrelated regression. J. Amer. Statist. Assoc. 84 598-601.
  • SKOVGAARD, I. M. (1981). Edgeworth expansions of the distributions of maximum likelihood estimators in the general (non i.i.d.) case. Scand. J. Statist. 8 227-236.
  • SKOVGAARD, I. M. (1986). On multivariate Edgeworth expansions. Internat. Statist. Rev. 54 169- 186.
  • TANIGUCHI, M. (1983). On the second order asy mptotic efficiency of estimators of Gaussian ARMA processes. Ann. Statist. 11 157-169.
  • TANIGUCHI, M. (1984). Validity of Edgeworth expansions for statistics of time series. J. Time Ser. Anal. 5 37-51.
  • TANIGUCHI, M. (1986). Third order asy mptotic properties of maximum likelihood estimators for Gaussian ARMA processes. J. Multivariate Anal. 18 1-31.
  • TANIGUCHI, M. (1988). Asy mptotic expansions of the distributions of some test statistics for Gaussian ARMA processes. J. Multivariate Anal. 27 494-511.
  • TANIGUCHI, M. (1991). Higher Order Asy mptotic Theory for Time Series Analy sis. Lecture Notes in Statist. 68. Springer, Berlin.
  • WHITTLE, P. (1953). Estimation and information in stationary time series. Ark. Mat. 2 423-434.