Abstract
In this paper we study the Maximum Likelihood Estimator (MLE) of the vector parameter of an autoregressive process of order p with regular stationary Gaussian noise. We prove the large sample asymptotic properties of the MLE under very mild conditions. We do simulations for fractional Gaussian noise (fGn), autoregressive noise (AR(1)) and moving average noise (MA(1)).
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References
J. Anděl, “Long Memory Time Series Models”, Kybernetika 22(2), 105–123 (1986).
T. W. Anderson, “On Asymptotic Distribution of Estimates of Parameters of Stochastic Difference Equations”, Ann. Math. Statist. 30, 676–687 (1959).
A. Brouste and M. Kleptsyna, “Kalman’s Type Filter under Stationary Noises”, Systems and Control Letters, No. 61, 1229–1234 (2012).
N. H. Chan and C. Z. Wei, “Asymptotic Inference for Nearly Nonstationary AR(1) Processes”, Ann. Statist. 15(3), 1050–1063 (1987).
M. Duflo, R. Senoussi, and A. Touati, “Propriétés asymptotiques presque süres de l’estimateur des moindres carrés d’un modèle autorégressif vectoriel”, Ann. Inst. H. Poincaré 27(1), 1–25 (1991).
J. Durbin, “The Fitting of Time Series Models”, Rev. Inst. Intern. Statist. 28, 233–243 (1960).
R. Fox and M. Taqqu, “Large-Sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time Series”, Ann. Statist. 14(2), 517–532 (1986).
C. W. Granger and R. Joyeux, “An Introduction to Long-Memory Time Series Models and Fractional Differencing”, J. Time Series Anal. 1(1), 15–29 (1980).
M. T. Greene and B. D. Fielitz, “Long-Term Dependence in Common Stock Returns”, J. Financial Economics 4, 339–349 (1977).
J. R. M. Hosking, “Fractional Differencing”, Biometrika 68(1), 165–176 (1981).
M. Kleptsyna, A. Le Breton, and M. Viot, “New Formulas Concerning Laplace Transforms of Quadratic Forms for General Gaussian Sequences”, J. Appl.Math. and Stoch. Anal., No. 1 (2002).
T. L. Lai and C. Z. Wei, “Asymptotic Properties of General Autoregressive Models and Strong Consistency of Least Square Estimate and Their Parameters”, J. Multivar.Anal. 13, 1–23 (1983).
A. J. Lawrance and N. T. Kottegoda, “Stochastic Modeling of River Flow Time Series”, J. Roy. Statist. Soc. 140, 1–27 (1977).
A. LeBretonand D. T. Pham, “On the Bias of the Least Squares Estimator for the First-Order Autoregressive Process”, Ann. Inst. Statist. Math. 41(3), 555–563 (1989).
R. Liptser and A. Shiryaev, Theory of Martingales (Kluwer Academic, 1989).
R. Liptser and A. Shiryaev, Statistics of Random Processes (Springer, 2001).
H. Mann and A. Wald, “On the Statistical Treatment of Linear Stochastic Difference Equations”, Econometrica 11(3–4), 173–220 (1943).
V. Maxim, L. Sendur, J. Fadili, J. Suckling, R. Gould, R. Howard, and E. Bullmore, “Fractional Gaussian Noise, Function MRI and Alzheimer’s Disease”, NeuroImage 25, 141–158 (2005).
M. M. Rao, Consistency and Limit Distributions of Estimators of Parameters in Explosive Stochastic Difference Equations”, Ann.Math. Statist. 32(1), 195–218 (1961).
M. M. Rao, “Asymptotic Distribution of an Estimator of the Boundary Parameter of an Unstable Process”, Ann. Statist. 6(1), 185–190 (1978).
H. Rubin, Consistency of Maximum Likelihood Estimates in the Explosive Case, in Statistical Inference in Dynamic Economic Models (Wiley and Sons, New York, 1950), pp. 356–364.
G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes (Chapman & Hall, 1994).
S. Sethuraman and I. V. Basawa, “The Asymptotic Distribution of the Maximum Likelihood Estimator for a Vector Time Series Model with Long Memory Dependence”, Statist. Probab. Lett. 31, 285–293 (1997).
Y. G. Sinai, “Self-similar Probability Distribution”, Theory Probab. Appl. 21(1), (1976).
T. J. Sweeting, “Uniform Asymptotic Normality of the Maximum Likelihood Estimator”, Ann. Statist. 8, 1375–1381 (1980).
J. White, “The Limiting Distribution of the Serial Correlation Coefficient in the ExplosiveCase I”, Ann.Math. Statist. 29, 1188–1197 (1958).
J. White, “The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case II”, Ann. Math. Statist. 30, 831–834 (1959).
P. Whittle, Hypothesis Testing in Time Series Analysis (Hafner, New York, 1951).
Y. Yajima, “On Estimation of Long-Memory Time Series Models”, Australian J. Statist. 27(3), 302–320 (1985).
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Brouste, A., Cai, C. & Kleptsyna, M. Asymptotic properties of the MLE for the autoregressive process coefficients under stationary Gaussian noise. Math. Meth. Stat. 23, 103–115 (2014). https://doi.org/10.3103/S1066530714020021
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DOI: https://doi.org/10.3103/S1066530714020021