Electronic Journal of Statistics

On the Bartlett correction of empirical likelihood for Gaussian long-memory time series

Ngai Hang Chan, Kun Chen, and Chun Yip Yau

Full-text: Open access


Bartlett correction is one of the desirable features of empirical likelihood (EL) since it allows constructions of confidence regions with improved coverage probabilities. Previous studies demonstrated the Bartlett correction of EL for independent observations and for short-memory time series. By establishing the validity of Edgeworth expansion for the signed root empirical log-likelihood ratio, the validity of Bartlett correction of EL for Gaussian long-memory time series is established. In particular, orders of the coverage error of confidence regions can be reduced from $\log^{6}n/n$ to $\log^{3}n/n$, which is different from the classical rate of reduction from $n^{-1}$ to $n^{-2}$.

Article information

Electron. J. Statist., Volume 8, Number 1 (2014), 1460-1490.

First available in Project Euclid: 26 August 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F10: Point estimation 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Coverage error Edgeworth expansion periodogram Whittle likelihood


Chan, Ngai Hang; Chen, Kun; Yau, Chun Yip. On the Bartlett correction of empirical likelihood for Gaussian long-memory time series. Electron. J. Statist. 8 (2014), no. 1, 1460--1490. doi:10.1214/14-EJS930. https://projecteuclid.org/euclid.ejs/1409058255

Export citation


  • [1] Andrews, D.W.K. and Liberman, O. (2005). Valid Edgeworth expansions for the whittle maximum likelihood estimator for stationary long-memory gaussian time series. Econometric Theory 21, 710–734.
  • [2] An, H.Z., Chen, Z.G. and Hannan, E.J. (1983). The maximum of the periodogram. Journal of Multivariate Analysis 13, 383–400.
  • [3] Barndorff-Nielsen, O.E. and Hall, P.(1988). On the level-error after Bartlett adjustment of the likelihood ratio statistic. Biometrika 75, 374–378.
  • [4] Beran, J. (1994) Statistics for Long-Memory Processes. Chapman & Hall, New York.
  • [5] Bhattacharya, R.N. and Ghosh, J.K. (1974). On the validity of the formal Edgeworth expansion. Annals of Statistics 6, 434–451.
  • [6] Brillinger, D.R. (1981). Time Series: Data Analysis and Theory. Holden-Day, San Francisco.
  • [7] Chambers, J.M. (1967). On methods of asymptotic approximation for multivariate distributions. Biometrika 54, 367–383.
  • [8] Chan, N.H. and Ling, S. (2006). Empirical likelihood for GARCH models. Econometric Theory 22, 402–428.
  • [9] Chan, N.H. and Liu, L. (2010). Bartlett correctability of empirical likelihood in time series. Journal of The Japanese Statistical Society 40, 1–5.
  • [10] Chen, X.S. and Cui, H. (2007). On the second order properties of empirical likelihood with moment restrictions. Journal of Econometrics 141, 492–516.
  • [11] Chen, X.S. and Hall, P. (1993). Smoothed empirical likelihood confidence intervals for quantiles. Annals of Statistics 21, 1166–1181.
  • [12] Chuang, C.S. and Chan, N.H. (2002). Empirical likelihood for autoregressive models with applications to unstable time series. Statistica Sinica 12, 387–407.
  • [13] Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Annals of Statistics 17, 1749–1766.
  • [14] Dahlhaus, R. and Janas D. (1996). A frequency domain bootstrap for ratio statistics in time series analysis. Annals of Statistics 24, 1934–1963.
  • [15] DiCiccio, T.J., Hall, P. and Romano, J.P. (1991). Empirical likelihood is Bartlett correctable. Annals of Statistics 19, 1053–1061.
  • [16] DiCiccio, T.J. and Romano, J.P. (1989). On adjustment based on the signed root of the empirical likelihood ratio statistics. Biometrika 76, 447–456.
  • [17] Granger, C.W.J. and Joyeux, R. (1980). An introduction to long-memory time series and fractional differencing. Journal of Time Series Analysis 1, 15–30.
  • [18] Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
  • [19] Hall, P. and La Scala, B. (1990). Methodology and algorithms of empirical likelihood. Internat. Statist. Rev. 58, 109–127.
  • [20] Hosking, J.R.M. (1981). Fractional differencing. Biometrika 68, 165–176.
  • [21] Hosoya, Y. (1997). A limit theory for long-range dependence and statistical inference on related models. Annals of Statistics 25, 105–137.
  • [22] Hurvich, C.M. and Beltrao K.I. (1993). Asymptotics for the low-frequency ordinates of the periodogram of a long-memory time series. Journal of Time Series Analysis 14, 455–472.
  • [23] Hurvich, C.M. and Zeger, S. (1987). Frequency domain bootstrap methods for time series. Statistics and Operations Research Working Paper, New York University, New York.
  • [24] Kitamura, Y. (1997). Empirical likelihood methods with weakly dependent processes. Annals of Statistics 25, 2084–2102.
  • [25] Kreiss, J.-P. and Paparoditis, E. (2011). Bootstrap for dependent data: a review (with discussion). Journal of the Korean Statistical Society 40, 357–395.
  • [26] Mandelbrot, B.B. and Van Ness, J.W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Review 10, 422–437.
  • [27] Monti, A.C. (1997). Empirical likelihood confidence regions in time series models. Biometrika 84, 395–405.
  • [28] Nordman, D.J. and Lahiri, S.N. (2006). A frequency domain empirical likelihood for short- and long-range dependence. Annals of Statistics 34, 3019–3050.
  • [29] Ogata, H. (2005). Empirical likelihood approach for non-Gaussian stationary processes. Scientiae Mathematicae Japonicae 62, 429–438.
  • [30] Owen, A.B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75, 237–249.
  • [31] Owen, A.B. (1990). Empirical likelihood ratio confidence regions. Annals of Statistics 18, 90–120.
  • [32] Owen, A.B. (2001). Empirical Likelihood. Chapman & Hall, New York.
  • [33] Priestley, M.B. (1981). Spectral Analysis and Time Series. Academic Press.
  • [34] Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating functions. Annals of Statistics 22, 300–325.
  • [35] Robinson, P.M. (1995). Log-periodogram regression of time series with long range dependence. Annals of Statistics 23, 1048–1072.
  • [36] Whittle, P. (1953). Estimation and information in stationary time series. Archiv. Math. 2, 423–434.
  • [37] Yau, C.Y. (2012). Empirical likelihood in long-memory time series models. Journal of Time Series Analysis 33, 269–275.
  • [38] Zhang, B. (1996). On the accuracy of empirical likelihood confidence intervals for M-Functionals. Journal of Nonparametric Statistics 6, 311–321.
  • [39] Zygmund, A. (1977). Trigonometric Series. Cambridge Univ. Press.