The Annals of Statistics

Local asymptotic normality for regression models with long-memory disturbance

Kokyo Choy, Marc Hallin, Abdeslam Serroukh, and Masanobu Taniguchi

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The local asymptotic normality property is established for a regression model with fractional ARIMA($p, d, q$) errors. This result allows for solving, in an asymptotically optimal way, a variety of inference problems in the long-memory context: hypothesis testing, discriminant analysis, rank-based testing, locally asymptotically minimax andadaptive estimation, etc. The problem of testing linear constraints on the parameters, the discriminant analysis problem, and the construction of locally asymptotically minimax adaptive estimators are treated in some detail.

Article information

Ann. Statist., Volume 27, Number 6 (1999), 2054-2080.

First available in Project Euclid: 4 April 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G10: Stationary processes 62E20: Asymptotic distribution theory
Secondary: 62A10 62F05: Asymptotic properties of tests

Long-memory process FARIMA model local asymptotic normality locally asymptotically optimal test adaptive estimation discriminant analysis


Hallin, Marc; Taniguchi, Masanobu; Serroukh, Abdeslam; Choy, Kokyo. Local asymptotic normality for regression models with long-memory disturbance. Ann. Statist. 27 (1999), no. 6, 2054--2080. doi:10.1214/aos/1017939250.

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  • Ash, R. B. (1972). Real Analysis and Probability. Academic Press, New York.
  • Benghabrit, Y. and Hallin, M. (1998). Locally asymptotically optimal tests for AR p against diagonal bilinear dependence. J. Statist. Plann. Inference 68 47-63.
  • Beran, J. (1994). Statistics for Long-Memory Processes. Chapman andHall, New York.
  • Beran, J. (1995). Maximum likelihoodof the differencing parameter for invertible short andlong memory autoregressive integratedmoving average models. J. Roy. Statist. Soc. Ser. B 57 659-672.
  • Bickel, P. J. and Klaassen, C. A. J. (1986). Empirical Bayes estimation in functional andstructural models, and uniformly adaptive estimation of location. Adv. in Appl. Math. 7 55-69.
  • Brillinger, D. R. (1981). Time Series. Data Analysis and Theory, 2nd ed. Holden-Day, San Francisco.
  • Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17 1749-1766.
  • Drost, F. C., Klaassen, C. A. J. and Werker, B. J. M. (1997). Adaptive estimation in time series models. Ann. Statist. 25 786-818.
  • Fuller, W. A. (1996). Introduction to Statistical Time Series, 2nded. Wiley, New York.
  • Garel, B. and Hallin, M. (1995). Local asymptotic normality of multivariate ARMA processes with linear trend. Ann. Inst. Statist. Math. 47 551-579.
  • Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle's estimate. Probab. Theory Related Fields 86 87-104.
  • Hallin, M. and Puri, M. L. (1988). Optimal rank-basedprocedures for time series analysis: testing an ARMA model against other ARMA models. Ann. Statist. 16 402-432.
  • Hallin, M. and Puri, M. L. (1994). Alignedrank tests for linear models with autocorrelatederror terms. J. Multivariate Anal. 50 175-237.
  • Hallin, M. and Serroukh, A. (1999). Adaptive estimation of the lag of a long-memory process. Statist. Inference Stochastic Process. 2 1-19.
  • Hallin, M. and Werker, B. (1998). Optimal testing for semi-parametric AR models: from Lagrange multipliers to autoregression rank scores andadaptive tests. In Asymptotics, Nonparametrics and Time Series (S. Ghosh, ed.) 295-358 M. Dekker, New York.
  • Hannan, E. J. (1970). Multiple Time Series. Wiley, New York.
  • Hwang, S.Y. and Basawa, I. V. (1993). Asymptotic optimal inference for a class of nonlinear time series models. Stochastic Process. Appl. 46 91-113.
  • Jeganathan, P. (1995). Some aspects of asymptotic theory with applications to time-series models. Econometric Theory 11 818-887.
  • Jeganathan, P. (1997). On asymptotic inference in linear cointegratedtime-series systems. Econometric Theory 13 692-745.
  • Koul, H. L. and Schick, A. (1997). Efficient estimation in nonlinear autoregressive time-series models. Bernoulli 3 247-277.
  • Kreiss, J. P. (1987). On adaptive estimation in stationary ARMA processes. Ann. Statist. 15 112-133. Kreiss, J. P. (1990a). Testing linear hypotheses in autoregression. Ann. of Statist. 18 1470-1482. Kreiss, J. P. (1990b). Local asymptotic normality for autoregression with infinite order. J. Statist. Plann. Inference 26 185-219.
  • Le Cam, L. (1960). Locally asymptotically normal families of distributions. University of California Publications in Statistics 3 37-98.
  • Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York.
  • Le Cam, L. and Yang, G. L. (1990). Asymptotics in Statistics. Springer, New York.
  • McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. Ann. Probab. 4 620-633.
  • Schick, A. (1993). On efficient estimation in regression models. Ann. Statist. 21 1486-1521.
  • Sethuraman, S. and Basawa, I. V. (1997). The asymptotic distribution of the maximum likelihood estimator for a vector time series model with long memory dependence. Statist. Probab. Letters 31 285-293.
  • Serroukh, A. (1996). Inf´erence asymptotique param´etrique et non param´etrique pour les mod eles ARMA fractionnaires. Ph.D. thesis, Institut Statist., Univ. Libre Bruxelles.
  • Shumway, R. H. and Unger, A. N. (1974). Linear discriminant functions for stationary time series. J. Amer. Statist. Assoc. 69 948-956.
  • Strasser, H. (1985). Mathematical Theory of Statistics. W. de Gruyter, Berlin.
  • Swensen, A. R. (1985). The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend. J. Multivariate Anal. 16 54-70.
  • Taniguchi, M. (1998). Statistical analysis basedon functionals of nonparametric spectral density estimators. In Asymptotics, Nonparametrics and Time Series (S. Ghosh, ed.) 351-394 M. Dekker, New York.
  • Yajima, Y. (1985). On estimation of long memory time series models. Austral. J. Statist. 27 303- 320.
  • Yajima, Y. (1991). Asymptotic properties of the LSE in a regression model with long memory stationary errors. Ann. Statist. 19 158-177.
  • Zhang, G. and Taniguchi, M. (1994). Discriminant analysis for stationary vector time series. J. Time Ser. Anal. 15 117-126.