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Estimating the order of a hidden Markov model. (English) Zbl 1018.62062

Summary: While the estimation of the parameters of a hidden Markov model has been studied extensively, the consistent estimation of the number of hidden states is still an unsolved problem. The AIC and BIC methods are used most commonly, but their use in this context has not been justified theoretically. The author shows that for many common models, the penalized minimum-distance method yields a consistent estimate of the number of hidden states in a stationary hidden Markov model. In addition to addressing the identifiability issues, she applies her method to a multiple sclerosis data set and assesses its performance via simulation.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62F10 Point estimation
62F12 Asymptotic properties of parametric estimators
62M09 Non-Markovian processes: estimation
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References:

[1] Albert, Time series for modelling counts from a relapsing-remitting disease: application to modelling disease activity in multiple sclerosis, Statistics in Medicine 13 pp 453– (1994)
[2] Baras, Consistent estimation of the order of hidden Markov chains pp 26– (1992) · Zbl 0784.60071
[3] Bickel, Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models, The Annals of Statistics 26 pp 1614– (1998) · Zbl 0932.62097
[4] Billingsley, Probability and Measure (1995)
[5] Chen, Penalized minimum-distance estimates in finite mixture models, The Canadian Journal of Statistics 24 pp 167– (1996) · Zbl 0858.62019
[6] Doukhan, Mixing: Properties and Examples (1994)
[7] Dortet-Bernadet, Choix de rnodèle pour des chaines de Markov cachées, Comptes rendus de I’Acad’emie des sciences, Série I: Mathématique 332 pp 469– (2001)
[8] Hettmansperger, Almost nonparametric inference for repeated measures in mixture models, Journal of the Royal Statistical Society Series B 62 pp 811– (2000) · Zbl 0957.62026
[9] Hughes, A class of stochastic models for relating synoptic atmospheric patterns to regional hydrologic phenomena, Water Resources Research 30 pp 1535– (1994)
[10] Ibragimov, Independent and Stationary Sequences of Random Variables (1971)
[11] Kieffer, Strongly consistent code-based identification and order estimation for constrained finite-state model classes, IEEE Transactions on Information Theory 39 pp 893– (1993) · Zbl 0784.94005
[12] Leroux, Maximum-likelihood estimation for hidden Markov models, Stochastic Processes and Their Applications 40 pp 127– (1992) · Zbl 0738.62081
[13] Leroux, Consistent estimation of a mixing distribution, The Annals of Statistics 20 pp 1350– (1992) · Zbl 0763.62015
[14] Leroux, Maximum-penalized likelihood estimation for independent and Markov-dependent mixture models, Biometrics 48 pp 545– (1992)
[15] Levinson, An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition, The Bell System Technical Journal 62 pp 1035– (1983) · Zbl 0507.68058 · doi:10.1002/j.1538-7305.1983.tb03114.x
[16] Lindgren, Markov regime models for mixed distributions and switching regressions, Scandinavian Journal of Statistics 5 pp 81– (1978) · Zbl 0382.62073
[17] Liu, Order estimation and sequential universal data compression of a hidden Markov source by the method of mixtures, IEEE Transactions on Information Theory 40 pp 1167– (1994) · Zbl 0811.94022
[18] MacDonald, Hidden Markov Models and Other Models for Discrete-Valued Time Series (1997) · Zbl 0868.60036
[19] Nash, Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation (1979) · Zbl 0697.68004
[20] Child-Saïd, Loi du log itéré pour la fonction de répartition erapirique dans le cas multidimensionel et {\(\alpha\)}-mélangeant, Comptes rendus de I’Académie des sciences, Série I: Mathe’matique 318 pp 759– (1994)
[21] Petrie, Probabilistic functions of finite state Markov chains, The Annals of Mathematical Statistics 40 pp 97– (1969) · Zbl 0181.21201
[22] Poskitt, Markov chain models, time series analysis and extreme value theory, Advances in Applied Probability 28 pp 405– (1996) · Zbl 0853.62068
[23] Prakasa Rao, Identifiability in Stochastic Models: Characterization of Probability Distributions (1992) · Zbl 0746.62008
[24] Robert, Bayesian inference in hidden Markov models through the reversible jump Markov chain Monte Carlo method, Journal of the Royal Statistical Society Series B 62 pp 57– (2000) · Zbl 0941.62090
[25] Rydén, Estimating the order of hidden Markov models, Statistics 26 pp 345– (1995) · Zbl 0836.62057
[26] Strichartz, The Way of Analysis (1995) · Zbl 0878.26001
[27] Teicher, Identifiability of mixtures of product measures, The Annals of Mathematical Statistics 38 pp 1300– (1967) · Zbl 0153.47904
[28] Wang, Markov Poisson regression models for discrete time series, Journal of Applied Statistics 26 pp 855– (1999) · Zbl 1008.62089
[29] White, Lumpable hidden Markov models-model reduction and reduced complexity filtering, IEEE Transactions on Automatic Control 45 pp 2297– (2000) · Zbl 0991.93114
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