## 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.

### MSC:

 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)
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### References:

 [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
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