We focus on the parametric estimation of the distribution of a Markov environment from the observation of a single trajectory of a one-dimensional nearest-neighbor path evolving in this random environment. In the ballistic case, as the length of the path increases, we prove consistency, asymptotic normality and efficiency of the maximum likelihood estimator. Our contribution is two-fold: we cast the problem into the one of parameter estimation in a hidden Markov model (HMM) and establish that the bivariate Markov chain underlying this HMM is positive Harris recurrent. We provide different examples of setups in which our results apply, in particular that of DNA unzipping model, and we give a simple synthetic experiment to illustrate those results.
DOI : 10.1051/ps/2015008
Mots clés : Hidden Markov model, Markov environment, maximum likelihood estimation, random walk in random environment
@article{PS_2015__19__605_0,
author = {Andreoletti, Pierre and Loukianova, Dasha and Matias, Catherine},
title = {Hidden {Markov} model for parameter estimation of a random walk in a {Markov} environment},
journal = {ESAIM: Probability and Statistics},
pages = {605--625},
publisher = {EDP-Sciences},
volume = {19},
year = {2015},
doi = {10.1051/ps/2015008},
mrnumber = {3433429},
zbl = {1392.62246},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2015008/}
}
TY - JOUR AU - Andreoletti, Pierre AU - Loukianova, Dasha AU - Matias, Catherine TI - Hidden Markov model for parameter estimation of a random walk in a Markov environment JO - ESAIM: Probability and Statistics PY - 2015 SP - 605 EP - 625 VL - 19 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2015008/ DO - 10.1051/ps/2015008 LA - en ID - PS_2015__19__605_0 ER -
%0 Journal Article %A Andreoletti, Pierre %A Loukianova, Dasha %A Matias, Catherine %T Hidden Markov model for parameter estimation of a random walk in a Markov environment %J ESAIM: Probability and Statistics %D 2015 %P 605-625 %V 19 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2015008/ %R 10.1051/ps/2015008 %G en %F PS_2015__19__605_0
Andreoletti, Pierre; Loukianova, Dasha; Matias, Catherine. Hidden Markov model for parameter estimation of a random walk in a Markov environment. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 605-625. doi : 10.1051/ps/2015008. http://www.numdam.org/articles/10.1051/ps/2015008/
and , Random walks in random environment: what a single trajectory tells. Israel J. Math. 142 (2004) 205–220. | MR | Zbl
. Asymptotic behaviour for random walks in random environments. J. Appl. Probab. 36 (1999) 334–349. | MR | Zbl
. On the concentration of Sinai’s walk. Stochastic Process. Appl. 116 (2006) 1377–1408. | MR | Zbl
, Almost sure estimates for the concentration neighborhood of Sinai’s walk. Stochastic Processes Appl. 117 (2007) 1473–1490. | MR | Zbl
, On the estimation of the potential of Sinai’s RWRE. Braz. J. Probab. Stat. 25 (2011) 121–144. | MR | Zbl
and , DNA unzipping via stopped birth and death processes with unknown transition probabilities. Appl. Math. Res. eXpress 2012 (2012) 184–208. | MR | Zbl
, , and , Inference of DNA sequences from mechanical unzipping: an ideal-case study. Phys. Rev. Lett. 96 (2006) 128–102.
and , Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Statist. 37 (1966) 1554–1563. | MR | Zbl
, , and , A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Statist. 41 (1970) 164–171. | MR | Zbl
and , Inference in hidden Markov models. I: Local asymptotic normality in the stationary case. Bernoulli 2 (1996) 199–228. | MR | Zbl
, and , Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models. Ann. Stat. 26 (1998) 1614–1635. | MR | Zbl
L. Bogachev, Random walks in random environments. Edited by J.P. Francoise, G. Naber and S.T. Tsou. Encycl. Math. Phys. 4 (2006) 353–371. | MR
, , and , A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16 (1995) 1190–1208. | MR | Zbl
O. Cappé, E. Moulines and T. Rydén, Inference in hidden Markov models. Springer Ser. Statist. Springer, New York (2005). | MR | Zbl
and , Number of hidden states and memory: a joint order estimation problem for Markov chains with Markov regime. ESAIM: PS 13 (2009) 38–50. | MR | Zbl
, Replication of a multicomponent chain by the lightning mechanism. Biofizika 12 (1967) 297–301.
F. Comets, M. Falconnet, O. Loukianov and D. Loukianova, Maximum likelihood estimator consistency for recurrent random walk in a parametric random environment with finite support. Technical report. Preprint (2014). | arXiv | MR
, , , and , Maximum likelihood estimator consistency for ballistic random walk in a parametric random environment. Stochastic Processes Appl. 124 (2014) 268–288. | MR | Zbl
, and , Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B 39 (1977) 1–38. | MR | Zbl
and , Asymptotics of the maximum likelihood estimator for general hidden Markov models. Bernoulli 7 (2001) 381–420. | MR | Zbl
, , and , Consistency of the maximum likelihood estimator for general hidden Markov models. Ann. Stat. 39 (2011) 474–513. | MR | Zbl
, and , Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Ann. Stat. 32 (2004) 2254–2304. | MR | Zbl
and , Hidden Markov processes. Special issue on Shannon theory: perspective, trends, and applications. IEEE Trans. Inform. Theory 48 (2002) 1518–1569. | MR | Zbl
, and , Maximum likelihood estimation in the context of a sub-ballistic random walk in a parametric random environment. Math. Methods Stat. 23 (2014) 159–175. | MR | Zbl
, and , Asymptotic normality and efficiency of the maximum likelihood estimator for the parameter of a ballistic random walk in a random environment. Math. Methods Stat. 23 (2014) 1–19. | MR | Zbl
and , Leroux’s method for general hidden Markov models. Stochastic Processes Appl. 116 (2006) 222–243. | MR | Zbl
B.D. Hughes, Random walks and random environments, Random environments. Vol. 2 of Oxford Science Publications. The Clarendon Press Oxford University Press, New York (1996). | MR | Zbl
and , Asymptotic normality of the maximum likelihood estimator in state space models. Ann. Statist. 27 (1999) 514–535. | MR | Zbl
, and , A limit law for random walk in a random environment. Compos. Math. 30 (1975) 145–168. | MR | Zbl
and , Exponential forgetting and geometric ergodicity in hidden Markov models. Math. Control Signals Syst. 13 (2000) 63–93. | MR | Zbl
, Maximum-likelihood estimation for hidden Markov models. Stochastic Process. Appl. 40 (1992) 127–143. | MR | Zbl
, Finding the observed information matrix when using the EM algorithm. J. R. Stat. Soc. Ser. B 44 (1982) 226–233. | MR | Zbl
S. Meyn and R.L. Tweedie, Markov chains and stochastic stability, 2nd edition. Cambridge University Press, Cambridge (2009). | MR | Zbl
, Reversibility of first-order autoregressive processes. Stochastic Processes Appl. 28 (1988) 61–69. | MR | Zbl
P. Révész, Random walk in random and non-random environments, 2nd edition. World Scientific (2005). | MR
, Sinai’s walk via stochastic calculus. Panoramas et Synthèses 12 (2001) 53–74. | MR | Zbl
, The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab. Appl. 27 (1982) 247–258. | MR | Zbl
, Random walks in a random environment. Ann. Probab. 3 (1975) 1–31. | MR | Zbl
, One-dimensional random walks in a two-component chain. Soviet Math. Dokl. 13 (1972) 1172–1176. | MR | Zbl
O. Zeitouni, Random walks in random environment. In Lectures on probability theory and statistics. Vol. 1837 of Lect. Notes Math. Springer, Berlin (2004) 189–312. | MR | Zbl
Cité par Sources :
