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.
Accepté le :
DOI : 10.1051/ps/2018017
Mots-clés : Inhomogeneous hidden Markov models, quasi-maximum likelihood estimation, strong consistency, robustness, asymptotic mean stationarity
@article{PS_2019__23__492_0,
author = {Diehn, Manuel and Munk, Axel and Rudolf, Daniel},
title = {Maximum likelihood estimation in hidden {Markov} models with inhomogeneous noise},
journal = {ESAIM: Probability and Statistics},
pages = {492--523},
publisher = {EDP-Sciences},
volume = {23},
year = {2019},
doi = {10.1051/ps/2018017},
mrnumber = {3989601},
zbl = {1422.62099},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2018017/}
}
TY - JOUR AU - Diehn, Manuel AU - Munk, Axel AU - Rudolf, Daniel TI - Maximum likelihood estimation in hidden Markov models with inhomogeneous noise JO - ESAIM: Probability and Statistics PY - 2019 SP - 492 EP - 523 VL - 23 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2018017/ DO - 10.1051/ps/2018017 LA - en ID - PS_2019__23__492_0 ER -
%0 Journal Article %A Diehn, Manuel %A Munk, Axel %A Rudolf, Daniel %T Maximum likelihood estimation in hidden Markov models with inhomogeneous noise %J ESAIM: Probability and Statistics %D 2019 %P 492-523 %V 23 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2018017/ %R 10.1051/ps/2018017 %G en %F PS_2019__23__492_0
Diehn, Manuel; Munk, Axel; Rudolf, Daniel. Maximum likelihood estimation in hidden Markov models with inhomogeneous noise. ESAIM: Probability and Statistics, Tome 23 (2019), pp. 492-523. doi : 10.1051/ps/2018017. http://archive.numdam.org/articles/10.1051/ps/2018017/
[1] and , Consistency of the maximum likelihood estimate for non-homogeneous Markov-switching models. ESAIM: PS 19 (2015) 268–292. | DOI | Numdam | MR | Zbl
[2] , The strong ergodic theorem for densities: generalized Shannon-McMillan-Breiman theorem. Ann. Probab. 13 (1985) 1292–1303. | DOI | MR | Zbl
[3] and , Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Statist. 37 (1966) 1554–1563. | DOI | MR | Zbl
[4] , , and , A maximization technique occuring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Statist. 41 (1970) 164–171. | DOI | MR | Zbl
[5] , and , Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models. Ann. Statist. 26 (1998) 1614–1635. | DOI | MR | Zbl
[6] , Convergence of Probability Measures, 2nd edition. Wiley-, New York (1999). | DOI | MR | Zbl
[7] , , , , , , , , , , and , Voltage dependence of conformational dynamics and subconducting states of VDAC-1. Biophys. J. 111 (2016) 1223–1234. | DOI
[8] , , , and , Interaction of zwitterionic penicillins with the OmpF channel facilitates their translocation. Biophys. J. 90 (2006) 1617–1627. | DOI
[9] and , Asymptotics of the maximum likelihood estimator for general hidden Markov models. Bernoulli 7 (2001) 381–420. | DOI | MR | Zbl
[10] , and , Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Ann. Statist. 32 (2004) 2254–2304. | DOI | MR | Zbl
[11] , , and , Consistency of the maximum likelihood estimator for general hidden Markov models. Ann. Statist. 39 (2011) 474–513. | DOI | MR | Zbl
[12] and , Leroux’s method for general hidden Markov models. Stoch. Process. Their Appl. 116 (2006) 222–243. | DOI | MR | Zbl
[13] and , Asymptotically mean stationary measures. Ann. Probab. 8 (1980) 962–973. | MR | Zbl
[14] , Probability, Random Processes, and Ergodic Properties, 2nd edition. Springer Publishing Company, Incorporated, New York (2009). | DOI | MR | Zbl
[15] and , Consistent maximum-likelihood estimation with dependent observations: the general (non-normal) case. J. Econom. 32 (1986) 253–285. | DOI | MR | Zbl
[16] , , , , , and , Idealizing ion channel recordings by a jump segmentation multiresolution filter. IEEE Trans. Nanobiosci. 12 (2013) 376–386. | DOI
[17] , Asymptotic normality of M-estimators in nonhomogeneous hidden Markov models. J. Appl. Probab. 48A (2011) 295–306. | DOI | MR | Zbl
[18] and , Asymptotic properties of the MLE in hidden Markov models, in 1997 European Control Conference (ECC) (1997) 3440–3445. | DOI
[19] , Maximum-likelihood estimation for hidden Markov models. Stoch. Process. Their Appl. 40 (1992) 127–143. | DOI | MR | Zbl
[20] and , The patch clamp technique. Sci. Am. 266 (1992) 44–51. | DOI
[21] , and , Maximum likelihood estimation in possibly misspecified dynamic models with time inhomogeneous Markov regimes, SSRN Scholarly. Social Science Research Network, Rochester, NY (2016) 2887771.
[22] , and , Hidden Markov modeling for single channel kinetics with filtering and correlated noise. Biophys. J. 79 (2000) 1928–1944. | DOI
[23] , Invariant measures for many-one transformations. Duke Math. J. 23 (1956) 477–488. | DOI | MR | Zbl
[24] , , , , , and , MCMC estimation of Markov models for ion channels. Biophys. J. 100 (2011) 1919–1929. | DOI
[25] , A new algorithm for idealizing single ion channel data containing multiple unknown conductance levels. Biophys. J. 70 (1996) 1303–1315. | DOI
[26] and , Applying hidden Markov models to the analysis of single ion channel activity. Biophys. J. 82 (2002) 1930–1942. | DOI
[27] , Note on the consistency of the maximum likelihood estimate. Ann. Math. Statist. 20 (1949) 595–601. | DOI | MR | Zbl
Cité par Sources :
