Consistency of the maximum likelihood estimator for general hidden Markov models. (English) Zbl 1209.62194

Summary: Consider a parametrized family of general hidden Markov models, where both the observed and unobserved components take values in a complete separable metric space. We prove that the maximum likelihood estimator (MLE) of the parameter is strongly consistent under a rather minimal set of assumptions. As special cases of our main result, we obtain consistency in a large class of nonlinear state space models, as well as general results on linear Gaussian state space models and finite state models.
A novel aspect of our approach is an information-theoretic technique for proving identifiability, which does not require an explicit representation for the relative entropy rate. Our method of proof could therefore form a foundation for the investigation of MLE consistency in more general dependent and non-Markovian time series. Also of independent interest is a general concentration inequality for \(V\)-uniformly ergodic Markov chains.


62M05 Markov processes: estimation; hidden Markov models
62F12 Asymptotic properties of parametric estimators
62B10 Statistical aspects of information-theoretic topics
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
Full Text: DOI arXiv


[1] Adamczak, R. (2008). A tail inequality for suprema of unbounded empirical processes with applications to Markov chains. Electron. J. Probab. 13 1000-1034. · Zbl 1190.60010
[2] Barron, A. (1985). The strong ergodic theorem for densities; generalized Shannon-McMillan-Breiman theorem. Ann. Probab. 13 1292-1303. · Zbl 0608.94001
[3] Baum, L. E. and Petrie, T. P. (1966). Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Statist. 37 1554-1563. · Zbl 0144.40902
[4] Bertsekas, D. P. and Shreve, S. E. (1978). Stochastic Optimal Control: The Discrete Time Case. Mathematics in Science and Engineering 139 . Academic Press, New York. · Zbl 0471.93002
[5] Billingsley, P. (1995). Probability and Measure , 3rd ed. Wiley, New York. · Zbl 0822.60002
[6] Cappé, O., Moulines, E. and Rydén, T. (2005). Inference in Hidden Markov Models . Springer, New York. · Zbl 1080.62065
[7] Churchill, G. (1992). Hidden Markov chains and the analysis of genome structure. Computers & Chemistry 16 107-115. · Zbl 0752.92015
[8] Douc, R. and Matias, C. (2001). Asymptotics of the maximum likelihood estimator for general hidden Markov models. Bernoulli 7 381-420. · Zbl 0987.62018
[9] Douc, R., Moulines, E. and Rydén, T. (2004). Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Ann. Statist. 32 2254-2304. · Zbl 1056.62028
[10] Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations . Wiley, New York. · Zbl 0904.60001
[11] Fredkin, D. and Rice, J. (1987). Correlation functions of a function of a finite-state Markov process with application to channel kinetics. Math. Biosci. 87 161-172. · Zbl 0632.92003
[12] Fuh, C.-D. (2006). Efficient likelihood estimation in state space models. Ann. Statist. 34 2026-2068. · Zbl 1373.62447
[13] Fuh, C.-D. (2010). Reply to “On some problems in the article Efficient Likelihood Estimation in State Space Models” by Cheng-Der Fuh [ Ann. Statist. 34 (2006) 2026-2068]. Ann. Statist. 38 1282-1285. · Zbl 1246.62185
[14] Genon-Catalot, V. and Laredo, C. (2006). Leroux’s method for general hidden Markov models. Stochastic Process. Appl. 116 222-243. · Zbl 1099.60022
[15] Glynn, P. W. and Meyn, S. P. (1996). A Liapounov bound for solutions of the Poisson equation. Ann. Probab. 24 916-931. · Zbl 0863.60063
[16] Glynn, P. W. and Ormoneit, D. (2002). Hoeffding’s inequality for uniformly ergodic Markov chains. Statist. Probab. Lett. 56 143-146. · Zbl 0999.60019
[17] Hull, J. and White, A. (1987). The pricing of options on assets with stochastic volatilities. J. Finance 42 281-300. · Zbl 1126.91369
[18] Jensen, J. L. (2010). On some problems in the article Efficient Likelihood Estimation in State Space Models by Cheng-Der Fuh [ Ann. Statist. 34 (2006) 2026-2068]. Ann. Statist. 38 1279-1281. · Zbl 1373.62448
[19] Juang, B. and Rabiner, L. (1991). Hidden Markov models for speech recognition. Technometrics 33 251-272. JSTOR: · Zbl 0762.62036
[20] Kalashnikov, V. V. (1994). Regeneration and general Markov chains. J. Appl. Math. Stochastic Anal. 7 357-371. · Zbl 0835.60059
[21] Le Gland, F. and Mevel, L. (2000). Basic properties of the projective product with application to products of column-allowable nonnegative matrices. Math. Control Signals Systems 13 41-62. · Zbl 0941.93012
[22] Le Gland, F. and Mevel, L. (2000). Exponential forgetting and geometric ergodicity in hidden Markov models. Math. Control Signals Systems 13 63-93. · Zbl 0941.93053
[23] Leroux, B. G. (1992). Maximum-likelihood estimation for hidden Markov models. Stochastic Process. Appl. 40 127-143. · Zbl 0738.62081
[24] Liebscher, E. (2005). Towards a unified approach for proving geometric ergodicity and mixing properties of nonlinear autoregressive processes. J. Time Ser. Anal. 26 669-689. · Zbl 1092.62091
[25] Mamon, R. S. and Elliott, R. J. (2007). Hidden Markov Models in Finance. International Series in Operations Research & Management Science 104 . Springer, Berlin. · Zbl 1116.91007
[26] Marton, K. and Shields, P. C. (1994). The positive-divergence and blowing-up properties. Israel J. Math. 86 331-348. · Zbl 0797.60044
[27] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability . Springer, London. · Zbl 0925.60001
[28] Petrie, T. (1969). Probabilistic functions of finite state Markov chains. Ann. Math. Statist. 40 97-115. · Zbl 0181.21201
[29] Roberts, G. O. and Tweedie, R. L. (1999). Bounds on regeneration times and convergence rates for Markov chains. Stochastic Process. Appl. 80 211-229. · Zbl 0961.60066
[30] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes . Springer, New York. · Zbl 0862.60002
[31] van Handel, R. (2009). The stability of conditional Markov processes and Markov chains in random environments. Ann. Probab. 37 1876-1925. · Zbl 1178.93142
[32] Williams, D. (1991). Probability With Martingales . Cambridge Univ. Press, Cambridge. · Zbl 0722.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.