We prove the consistency of the maximum likelihood estimator for a large family of models generalizing the well known Markov-switching AutoRegressive (MS-AR) models by letting the transition probabilities vary in time and depend on covariates. We illustrate our theoretical result on the famous MacKenzie River lynx dataset and on a multi-site model for downscaling rainfall.
DOI : 10.1051/ps/2014024
Mots clés : Markov-switching autoregressive process, non-homogeneous hidden Markov process, maximum likelihood, consistency, stability, lynx data
@article{PS_2015__19__268_0,
author = {Ailliot, Pierre and P\`ene, Fran\c{c}oise},
title = {Consistency of the maximum likelihood estimate for non-homogeneous {Markov{\textendash}switching} models},
journal = {ESAIM: Probability and Statistics},
pages = {268--292},
publisher = {EDP-Sciences},
volume = {19},
year = {2015},
doi = {10.1051/ps/2014024},
mrnumber = {3412646},
zbl = {1330.62101},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2014024/}
}
TY - JOUR AU - Ailliot, Pierre AU - Pène, Françoise TI - Consistency of the maximum likelihood estimate for non-homogeneous Markov–switching models JO - ESAIM: Probability and Statistics PY - 2015 SP - 268 EP - 292 VL - 19 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2014024/ DO - 10.1051/ps/2014024 LA - en ID - PS_2015__19__268_0 ER -
%0 Journal Article %A Ailliot, Pierre %A Pène, Françoise %T Consistency of the maximum likelihood estimate for non-homogeneous Markov–switching models %J ESAIM: Probability and Statistics %D 2015 %P 268-292 %V 19 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2014024/ %R 10.1051/ps/2014024 %G en %F PS_2015__19__268_0
Ailliot, Pierre; Pène, Françoise. Consistency of the maximum likelihood estimate for non-homogeneous Markov–switching models. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 268-292. doi : 10.1051/ps/2014024. http://www.numdam.org/articles/10.1051/ps/2014024/
, Some theoretical results on Markov-switching autoregressive models with gamma innovations. C. R. Acad. Sci. Ser. I 343 (2006) 271–274. | MR | Zbl
and , Markov-switching autoregressive models for wind time series. Environ. Model. Softw. 30 (2012) 92–101.
, , , , Non-homogeneous hidden Markov-switching models for wind time series. J. Stat. Plann. Inference 160 (2015) 75–88. | MR | Zbl
and , The geometrical ergodicity of nonlinear autoregressive models. Stat. Sin. 6 (1996) 943–956. | MR | Zbl
, and , A hidden Markov model for downscaling synoptic atmospheric patterns to precipitation amounts. Climate Res. 15 (2000) 1–12.
O. Cappé, E. Moulines and T. Rydén, Inference in Hidden Markov Models. Springer-Verlag, New York (2005). | MR | Zbl
and , On the ergodicity of tar(1) processes. Ann. Appl. Probab. 1 (1991) 613–634. | MR | Zbl
F. Diebold, J-H Lee and G. Weinbach, Regime Switching with Time-Varying Transition Probabilities. Oxford University Press, Oxford (1994).
and , Asymptotics of the maximum-likelihood estimator for general hidden Markov models. Bernoulli 7 (2001) 381–420. | MR | Zbl
, and , Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Ann. Stat. 32 (2004) 2254–2304. | MR | Zbl
J. Fan and Q. Yao, Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York (2003). | MR | Zbl
and , Ergodicity of autoregressive processes with Markov-switching and consistency of the maximum-likelihood estimator. Statistics 32 (1998) 151–173. | MR | Zbl
, A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57 (1989) 357–384. | MR | Zbl
, and , A non-homogeneous hidden Markov model for precipitation occurrence. J. R. Stat. Soc. Series C 48 (1999) 15–30. | Zbl
, and , Estimation of Markov regime-switching regression models with endogenous switching. J. Econ. 143 (2008) 263–273. | MR | Zbl
and , Consistent estimation of linear and non-linear autoregressive models with Markov regime. J. Time Ser. Anal. 19 (1998) 291–307. | MR | Zbl
, Markov-switching vector autoregressions: modelling, statistical inference, and application to business cycle analysis. Lect. Notes Econ. Math. Systems 454 (1997). | MR | Zbl
and , Exponential forgetting and geometric ergodicity in hidden Markov models. Math. Control, Signals Syst. 13 (2000) 63–93. | MR | Zbl
, Maximum-likelihood estimation for hidden Markov models. Stoch. Proc. Appl. 40 (1992) 127–143. | MR | Zbl
T. Lindvall, Lectures on the Coupling Method. Corrected reprint of the 1992 original. Dover Publications, Inc., Mineola, NY (2002). | MR | Zbl
S.P. Meyn, R.L Tweedie and P.W. Glynn, Markov Chains and Stochastic Stability. Cambridge, Cambridge University Press (2009), vol. 2. | MR
, Identifiability of finite mixtures. Ann. Math. Stat. 34 (1963) 1265–1269. | MR | Zbl
H. Tong, Non-Linear Time Series: A Dynamical System Approach. Oxford, UK, Oxford University Press (1990). | MR | Zbl
, and , Confidence intervals for hidden Markov model parameters. British J. Math. Stat. Psychol. 53 (2000) 317–327.
and , Stochastic downscaling of precipitation: From dry events to heavy rainfalls. Water Res. Research 43 (2007) W07402.
, On square-integrability of an AR process with Markov switching. Stat. Probab. Lett. 52 (2001) 265–270. | MR | Zbl
and , On stability of nonlinear AR processes with Markov switching. Adv. Appl. Probab. 32 (2000) 394–407. | MR | Zbl
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