The Annals of Statistics
- Ann. Statist.
- Volume 32, Number 5 (2004), 2254-2304.
Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime
Randal Douc, Éric Moulines, and Tobias Rydén
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Abstract
An autoregressive process with Markov regime is an autoregressive process for which the regression function at each time point is given by a nonobservable Markov chain. In this paper we consider the asymptotic properties of the maximum likelihood estimator in a possibly nonstationary process of this kind for which the hidden state space is compact but not necessarily finite. Consistency and asymptotic normality are shown to follow from uniform exponential forgetting of the initial distribution for the hidden Markov chain conditional on the observations.
Article information
Source
Ann. Statist. Volume 32, Number 5 (2004), 2254-2304.
Dates
First available in Project Euclid: 27 October 2004
Permanent link to this document
http://projecteuclid.org/euclid.aos/1098883789
Digital Object Identifier
doi:10.1214/009053604000000021
Mathematical Reviews number (MathSciNet)
MR2102510
Zentralblatt MATH identifier
1056.62028
Subjects
Primary: 62M09: Non-Markovian processes: estimation
Secondary: 62F12: Asymptotic properties of estimators
Keywords
Asymptotic normality autoregressive process consistency geometric ergodicity hidden Markov model identifiability maximum likelihood switching autoregression
Citation
Douc, Randal; Moulines, Éric; Rydén, Tobias. Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Ann. Statist. 32 (2004), no. 5, 2254--2304. doi:10.1214/009053604000000021. http://projecteuclid.org/euclid.aos/1098883789.
References
- Bakry, D., Milhaud, X. and Vandekerkhove, P. (1997). Statistics of hidden Markov chains with finite state space. The nonstationary case. C. R. Acad. Sci. Paris Sér. I Math. 325 203--206. (In French.)Mathematical Reviews (MathSciNet): MR1467078
Digital Object Identifier: doi:10.1016/S0764-4442(97)84600-9 - Bar-Shalom, Y. and Li, X.-R. (1993). Estimation and Tracking. Principles, Techniques, and Software. Artech House, Boston.
- Baum, L. and Petrie, T. (1966). Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Statist. 37 1554--1563.Mathematical Reviews (MathSciNet): MR202264
Digital Object Identifier: doi:10.1214/aoms/1177699147
Project Euclid: euclid.aoms/1177699147 - Bickel, P. and Ritov, Y. (1996). Inference in hidden Markov models. I. Local asymptotic normality in the stationary case. Bernoulli 2 199--228.Mathematical Reviews (MathSciNet): MR1416863
Digital Object Identifier: doi:10.2307/3318520
Project Euclid: euclid.bj/1178291719
Zentralblatt MATH: 1066.62535 - Bickel, P., Ritov, Y. and Rydén, T. (1998). Asymptotic normality of the maximum likelihood estimator for general hidden Markov models. Ann. Statist. 26 1614--1635.Mathematical Reviews (MathSciNet): MR1647705
Digital Object Identifier: doi:10.1214/aos/1024691255
Project Euclid: euclid.aos/1024691255
Zentralblatt MATH: 0932.62097 - Booth, J. G. and Hobert, J. P. (1999). Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 265--285.
- Booth, J. G., Hobert, J. P. and Jank, W. (2001). A survey of Monte Carlo algorithms for maximizing the likelihood of a two-stage hierarchical model. Stat. Modelling 1 333--349.
- Cappé, O. (2001). Recursive computation of smoothed functionals of hidden Markovian processes using a particle approximation. Monte Carlo Methods Appl. 7 81--92.Mathematical Reviews (MathSciNet): MR1828199
- Chan, K. and Ledolter, J. (1995). Monte Carlo EM estimation for time series models involving counts. J. Amer. Statist. Assoc. 90 242--252.Mathematical Reviews (MathSciNet): MR1325132
Digital Object Identifier: doi:10.2307/2291149
JSTOR: links.jstor.org
Zentralblatt MATH: 0819.62069 - Chib, S., Nardari, F. and Shephard, N. (2002). Markov chain Monte Carlo methods for stochastic volatility models. J. Econometrics 108 281--316.Mathematical Reviews (MathSciNet): MR1894758
Digital Object Identifier: doi:10.1016/S0304-4076(01)00137-3
Zentralblatt MATH: 1099.62539 - Churchill, G. A. (1989). Stochastic models for heterogeneous DNA sequences. Bull. Math. Biol. 51 79--94.Mathematical Reviews (MathSciNet): MR978904
Digital Object Identifier: doi:10.1016/S0092-8240(89)80049-7
Zentralblatt MATH: 0662.92012 - de Jong, P. and Shephard, N. (1995). The simulation smoother for time series models. Biometrika 82 339--350.Mathematical Reviews (MathSciNet): MR1354233
Zentralblatt MATH: 0823.62072
Digital Object Identifier: doi:10.1093/biomet/82.2.339
JSTOR: links.jstor.org - Del Moral, P. and Guionnet, A. (2001). On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. H. Poincaré Probab. Statist. 37 155--194.Mathematical Reviews (MathSciNet): MR1819122
Digital Object Identifier: doi:10.1016/S0246-0203(00)01064-5 - Del Moral, P. and Miclo, L. (2000). Branching and interacting particle systems approximations of Feynman--Kac formulae with applications to non-linear filtering. Séminaire de Probabilités XXXIV. Lecture Notes in Math. 1729 1--145. Springer, Berlin.Mathematical Reviews (MathSciNet): MR1768060
- Del Moral, P. and Miclo, L. (2001). Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman--Kac semigroups. Technical report, Univ. Toulouse III.
- Douc, R. and Matias, C. (2001). Asymptotics of the maximum likelihood estimator for general hidden Markov models. Bernoulli 7 381--420.Mathematical Reviews (MathSciNet): MR1836737
Project Euclid: euclid.bj/1080004757
Digital Object Identifier: doi:10.2307/3318493
Zentralblatt MATH: 0987.62018 - Doucet, A., de Freitas, N. and Gordon, N. (2001). An introduction to sequential Monte Carlo methods. In Sequential Monte Carlo Methods in Practice (A. Doucet, N. de Freitas and N. Gordon, eds.) 3--14. Springer, New York.
- Doucet, A., Logothetis, A. and Krishnamurthy, V. (2000). Stochastic sampling algorithms for state estimation of jump Markov linear systems. IEEE Trans. Automat. Control 45 188--202.Mathematical Reviews (MathSciNet): MR1756516
Digital Object Identifier: doi:10.1109/9.839943
Zentralblatt MATH: 0980.93076 - Durrett, R. (1996). Probability: Theory and Examples, 2nd ed. Duxbury, Belmont, CA.Mathematical Reviews (MathSciNet): MR1609153
- Fort, G. and Moulines, E. (2003). Convergence of the Monte Carlo expectation maximization for curved exponential families. Ann. Statist. 31 1220--1259.Mathematical Reviews (MathSciNet): MR2001649
Digital Object Identifier: doi:10.1214/aos/1059655912
Project Euclid: euclid.aos/1059655912
Zentralblatt MATH: 1043.62015 - Francq, C. and Roussignol, M. (1998). Ergodicity of autoregressive processes with Markov-switching and consistency of the maximum-likelihood estimator. Statistics 32 151--173.Mathematical Reviews (MathSciNet): MR1708120
Digital Object Identifier: doi:10.1080/02331889808802659
Zentralblatt MATH: 0954.62104 - Francq, C. and Zakoian, J. M. (2001). Stationarity of multivariate Markov-switching ARMA models. J. Econometrics 102 339--364.Mathematical Reviews (MathSciNet): MR1842246
Digital Object Identifier: doi:10.1016/S0304-4076(01)00057-4
Zentralblatt MATH: 0998.62076 - Fredkin, D. R. and Rice, J. A. (1987). Correlation functions of a function of a finite-state Markov process with application to channel kinetics. Math. Biosci. 87 161--172.Mathematical Reviews (MathSciNet): MR929996
Digital Object Identifier: doi:10.1016/0025-5564(87)90072-1
Zentralblatt MATH: 0632.92003 - Geyer, G. J. (1994). On the convergence of Monte Carlo maximum likelihood computations. J. Roy. Statist. Soc. Ser. B 56 261--274.
- Geyer, G. J. and Thompson, E. A. (1992). Constrained Monte Carlo maximum likelihood for dependent data (with discussion). J. Roy. Statist. Soc. Ser. B 54 657--699.
- Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57 357--384.Mathematical Reviews (MathSciNet): MR996941
Digital Object Identifier: doi:10.2307/1912559
JSTOR: links.jstor.org
Zentralblatt MATH: 0685.62092 - Hamilton, J. D. (1990). Analysis of time series subject to changes in regime. J. Econometrics 45 39--70.Mathematical Reviews (MathSciNet): MR1067230
Digital Object Identifier: doi:10.1016/0304-4076(90)90093-9 - Holst, U., Lindgren, G., Holst, J. and Thuvesholmen, M. (1994). Recursive estimation in switching autoregressions with Markov regime. J. Time Series Anal. 15 489--506.Mathematical Reviews (MathSciNet): MR1292163
Digital Object Identifier: doi:10.1111/j.1467-9892.1994.tb00206.x
Zentralblatt MATH: 0807.62061 - Hürzeler, M. and Künsch, H. R. (2001). Approximating and maximising the likelihood for a general state space model. In Sequential Monte Carlo Methods in Practice (A. Doucet, N. de Freitas and N. Gordon, eds.) 159--175. Springer, New York.Mathematical Reviews (MathSciNet): MR1847791
- Jensen, J. L. and Petersen, N. V. (1999). Asymptotic normality of the maximum likelihood estimator in state space models. Ann. Statist. 27 514--535.Mathematical Reviews (MathSciNet): MR1714719
Digital Object Identifier: doi:10.1214/aos/1018031205
Project Euclid: euclid.aos/1018031205
Zentralblatt MATH: 0952.62023 - Juang, B.-H. and Rabiner, L. R. (1991). Hidden Markov models for speech recognition. Technometrics 33 251--272.Mathematical Reviews (MathSciNet): MR1132665
Digital Object Identifier: doi:10.2307/1268779
JSTOR: links.jstor.org
Zentralblatt MATH: 0762.62036 - Kim, C. and Nelson, C. (1999). State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications. MIT Press.
- Krishnamurthy, V. and Rydén, T. (1998). Consistent estimation of linear and non-linear autoregressive models with Markov regime. J. Time Series Anal. 19 291--307.
- Krolzig, H.-M. (1997). Markov-Switching Vector Autoregressions. Modelling, Statistical Inference, and Application to Business Cycle Analysis. Lecture Notes in Econom. and Math. Systems 454. Springer, Berlin.
- Künsch, H. R. (2001). State space and hidden Markov models. In Complex Stochastic Systems (O. E. Barndorff-Nielsen, D. R. Cox and C. Klüppelberg, eds.) 109--173. Chapman and Hall/CRC Press, Boca Raton, FL.Mathematical Reviews (MathSciNet): MR1893412
- Le Gland, F. and Mevel, L. (2000). Exponential forgetting and geometric ergodicity in hidden Markov models. Math. Control Signals Systems 13 63--93.Mathematical Reviews (MathSciNet): MR1742140
Digital Object Identifier: doi:10.1007/PL00009861
Zentralblatt MATH: 0941.93053 - Leroux, B. G. (1992). Maximum likelihood estimation for hidden Markov models. Stochastic Process. Appl. 40 127--143.Mathematical Reviews (MathSciNet): MR1145463
Digital Object Identifier: doi:10.1016/0304-4149(92)90141-C
Zentralblatt MATH: 0738.62081 - Lindvall, T. (1992). Lectures on the Coupling Method. Wiley, New York.
- Louis, T. A. (1982). Finding the observed information matrix when using the EM algorithm. J. Roy. Statist. Soc. Ser. B 44 226--233.
- MacDonald, I. L. and Zucchini, W. (1997). Hidden Markov and Other Models for Discrete-Valued Time Series. Chapman and Hall, London.
- Mann, H. B. and Wald, A. (1943). On the statistical treatment of linear stochastic difference equations. Econometrica 11 173--220.Mathematical Reviews (MathSciNet): MR9291
Digital Object Identifier: doi:10.2307/1905674
JSTOR: links.jstor.org
Zentralblatt MATH: 0063.03773 - Mevel, L. (1997). Statistique asymptotique pour les modèles de Markov cachés. Ph.D. thesis, Univ. Rennes 1.
- Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, Berlin.
- Ng, W., Larocque, J. and Reilly, J. (2001). On the implementation of particle filters for DOA tracking. In Proc. IEEE International Conference on Acoustics, Speech and Signal Processing 5 2821--2824. IEEE, Washington.
- Nielsen, S. F. (2000). The stochastic EM algorithm: Estimation and asymptotic results. Bernoulli 6 457--489.Mathematical Reviews (MathSciNet): MR1762556
Digital Object Identifier: doi:10.2307/3318671
Project Euclid: euclid.bj/1081616701
Zentralblatt MATH: 0981.62022 - Orton, M. and Fitzgerald, W. (2002). A Bayesian approach to tracking multiple targets using sensor arrays and particle filters. IEEE Trans. Signal Process. 50 216--223.
- Pitt, M. K. (2002). Smooth particle filters for likelihood evaluation and maximisation. Technical Report 651, Dept. Economics, Univ. Warwick.
- Shiryaev, A. N. (1996). Probability, 2nd ed. Springer, New York.Mathematical Reviews (MathSciNet): MR1368405
- Susmel, R. (2000). Switching volatility in private international equity markets. Internat. J. Finance Economics 5 265--283.
- Tanner, M. A. (1996). Tools for Statistical Inference. Methods for the Exploration of Posterior Distributions and Likelihood Functions, 3rd ed. Springer, New York.
- Thorisson, H. (2000). Coupling, Stationarity and Regeneration. Springer, New York.
- Tugnait, J. K. (1982). Adaptive estimation and identification for discrete systems with Markov jump parameters. IEEE Trans. Automat. Control 27 1054--1065. [Correction (1984) 29 286.]Mathematical Reviews (MathSciNet): MR696299
Digital Object Identifier: doi:10.1109/TAC.1982.1103061
Zentralblatt MATH: 0494.93045 - Wald, A. (1949). Note on the consistency of the maximum likelihood estimate. Ann. Math. Statist. 20 595--601.Mathematical Reviews (MathSciNet): MR32169
Digital Object Identifier: doi:10.1214/aoms/1177729952
Project Euclid: euclid.aoms/1177729952 - Yao, J.-F. and Attali, J.-G. (2000). On stability of nonlinear AR processes with Markov switching. Adv. in Appl. Probab. 32 394--407.Mathematical Reviews (MathSciNet): MR1778571
Digital Object Identifier: doi:10.1239/aap/1013540170
Project Euclid: euclid.aap/1013540170
Zentralblatt MATH: 0961.60076
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