×

Maximum likelihood estimation in hidden Markov models with inhomogeneous noise. (English) Zbl 1422.62099

Summary: We consider parameter estimation in finite hidden state space Markov models with time-dependent inhomogeneous noise, where the inhomogeneity vanishes sufficiently fast. Based on the concept of asymptotic mean stationary processes we prove that the maximum likelihood and a quasi-maximum likelihood estimator (QMLE) are strongly consistent. The computation of the QMLE ignores the inhomogeneity, hence, is much simpler and robust. The theory is motivated by an example from biophysics and applied to a Poisson- and linear Gaussian model.

MSC:

62F12 Asymptotic properties of parametric estimators
62M05 Markov processes: estimation; hidden Markov models
62F35 Robustness and adaptive procedures (parametric inference)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] P. Ailliot and F. Pène, Consistency of the maximum likelihood estimate for non-homogeneous Markov-switching models. ESAIM: PS19 (2015) 268-292. · Zbl 1330.62101 · doi:10.1051/ps/2014024
[2] A.R. Barron, The strong ergodic theorem for densities: generalized Shannon-McMillan-Breiman theorem. Ann. Probab. 13 (1985) 1292-1303. · Zbl 0608.94001
[3] L.E. Baum and T. Petrie, Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Statist. 37 (1966) 1554-1563. · Zbl 0144.40902 · doi:10.1214/aoms/1177699147
[4] L.E. Baum, T. Petrie, G. Soules and N. Weiss, A maximization technique occuring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Statist. 41 (1970) 164-171. · Zbl 0188.49603 · doi:10.1214/aoms/1177697196
[5] P.J. Bickel, Y. Ritov and T. Rydén, Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models. Ann. Statist. 26 (1998) 1614-1635. · Zbl 0932.62097 · doi:10.1214/aos/1024691255
[6] P. Billingsley, Convergence of Probability Measures, 2nd edition. Wiley-, New York (1999). · Zbl 0944.60003 · doi:10.1002/9780470316962
[7] R. Briones, C. Weichbrodt, L. Paltrinieri, I. Mey, S. Villinger, K. Giller, A. Lange, M. Zweckstetter, C. Griesinger, S. Becker, C. Steinem and B.L. de Groot, Voltage dependence of conformational dynamics and subconducting states of VDAC-1. Biophys. J. 111 (2016) 1223-1234. · doi:10.1016/j.bpj.2016.08.007
[8] C. Danelon, E.M. Nestorovich, M. Winterhalter, M. Ceccarelli and S.M. Bezrukov, Interaction of zwitterionic penicillins with the OmpF channel facilitates their translocation. Biophys. J. 90 (2006) 1617-1627. · doi:10.1529/biophysj.105.075192
[9] R. Douc and C. Matias, Asymptotics of the maximum likelihood estimator for general hidden Markov models. Bernoulli7 (2001) 381-420. · Zbl 0987.62018 · doi:10.2307/3318493
[10] R. Douc, E. Moulines and T. Ryden, Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Ann. Statist. 32 (2004) 2254-2304. · Zbl 1056.62028 · doi:10.1214/009053604000000021
[11] R. Douc, E. Moulines, J. Olsson and R. van Handel, Consistency of the maximum likelihood estimator for general hidden Markov models. Ann. Statist. 39 (2011) 474-513. · Zbl 1209.62194 · doi:10.1214/10-AOS834
[12] V. Genon-Catalot and C. Laredo, Leroux’s method for general hidden Markov models. Stoch. Process. Their Appl. 116 (2006) 222-243. · Zbl 1099.60022
[13] R.M. Gray and J.C. Kieffer, Asymptotically mean stationary measures. Ann. Probab. 8 (1980) 962-973. · Zbl 0447.28014
[14] R.M. Gray, Probability, Random Processes, and Ergodic Properties, 2nd edition. Springer Publishing Company, Incorporated, New York (2009). · Zbl 1191.60007 · doi:10.1007/978-1-4419-1090-5
[15] R.D.H. Heymans and J.R. Magnus, Consistent maximum-likelihood estimation with dependent observations: the general (non-normal) case. J. Econom. 32 (1986) 253-285. · Zbl 0659.62127
[16] T. Hotz, O.M. Schütte, H. Sieling, T. Polupanow, U. Diederichsen, C. Steinem and A. Munk, Idealizing ion channel recordings by a jump segmentation multiresolution filter. IEEE Trans. Nanobiosci.12 (2013) 376-386. · doi:10.1109/TNB.2013.2284063
[17] J.L. Jensen, Asymptotic normality of M-estimators in nonhomogeneous hidden Markov models. J. Appl. Probab. 48A (2011) 295-306. · Zbl 1230.62024
[18] F. LeGland and L. Mevel, Asymptotic properties of the MLE in hidden Markov models, in 1997 European Control Conference (ECC) (1997) 3440-3445. · doi:10.23919/ECC.1997.7082645
[19] B.G. Leroux, Maximum-likelihood estimation for hidden Markov models. Stoch. Process. Their Appl. 40 (1992) 127-143. · Zbl 0738.62081
[20] E. Neher and B. Sakmann, The patch clamp technique. Sci. Am. 266 (1992) 44-51. · doi:10.1038/scientificamerican0392-44
[21] D. Pouzo, Z. Psaradakis and M. Sola, Maximum likelihood estimation in possibly misspecified dynamic models with time inhomogeneous Markov regimes, SSRN Scholarly. Social Science Research Network, Rochester, NY (2016) 2887771.
[22] F. Qin, A. Auerbach and F. Sachs, Hidden Markov modeling for single channel kinetics with filtering and correlated noise. Biophys. J. 79 (2000) 1928-1944. · doi:10.1016/S0006-3495(00)76442-3
[23] O.W. Rechard, Invariant measures for many-one transformations. Duke Math. J. 23 (1956) 477-488. · Zbl 0070.28001 · doi:10.1215/S0012-7094-56-02344-4
[24] I. Siekmann, L.E. Wagner, D. Yule, C. Fox, D. Bryant, E.J. Crampin and J. Sneyd, MCMC estimation of Markov models for ion channels. Biophys. J. 100 (2011) 1919-1929. · doi:10.1016/j.bpj.2011.02.059
[25] A.M. VanDongen, A new algorithm for idealizing single ion channel data containing multiple unknown conductance levels. Biophys. J. 70 (1996) 1303-1315. · doi:10.1016/S0006-3495(96)79687-X
[26] L. Venkataramanan and F.J. Sigworth, Applying hidden Markov models to the analysis of single ion channel activity. Biophys. J. 82 (2002) 1930-1942. · doi:10.1016/S0006-3495(02)75542-2
[27] A. Wald, Note on the consistency of the maximum likelihood estimate. Ann. Math. Statist. 20 (1949) 595-601. · Zbl 0034.22902 · doi:10.1214/aoms/1177729952
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.