Abstract
We consider a one-dimensional sub-ballistic random walk evolving in a parametric i.i.d. random environment. We study the asymptotic properties of the maximum likelihood estimator (MLE) of the parameter based on a single observation of the path till the time it reaches a distant site. For that purpose, we adapt the method developed in the ballistic case by Comets et al. (2014) and Falconnet et al. (2014). Using a supplementary assumption due to the special nature of the sub-ballistic regime, we prove consistency and asymptotic normality as the distant site tends to infinity. To emphasize the role of the additional assumption, we investigate the Temkin model with unknown support, and it turns out that the MLE is consistent but, unlike the ballistic regime, the Fisher information is infinite. We also explore the numerical performance of our estimation procedure.
Similar content being viewed by others
References
O. Adelman and N. Enriquez, “Random Walks in Random Environment: What a Single Trajectory Tells”, Israel J. Math. 142, 205–220 (2004).
H. J. Bierens, Introduction to the Mathematical and Statistical Foundations of Econometrics, in Cambridge Books (Cambridge Univ. Press, Cambridge, 2005).
F. Comets, M. Falconnet, O. Loukianov, D. Loukianova, and C. Matias, “Maximum Likelihood Estimator Consistency for Ballistic RandomWalk in a Parametric Random Environment”, Stochastic Proc. Appl. 124, 268–288 (2014).
M. Falconnet, D. Loukianova, and C. Matias, “Asymptotic Normality and Efficiency of the Maximum Likelihood Estimator for the Parameter of a Ballistic Random Walk in a Random Environment”, Math. Methods Statist. 23(1), 1–19 (2014).
H. Kesten, “Random Difference Equations and Renewal Theory for Products of Random Matrices”, Acta Math. 131, 208–248 (1973).
H. Kesten, M. V. Kozlov, and F. Spitzer, “A Limit Law for Random Walk in a Random Environment”, Compositio Math. 30, 145–168 (1975).
D. Revuz, Markov chains, in North-Holland Mathematical Library (North-Holland, Amsterdam, 1984), Vol. 11, 2nd ed.
A. N. Shiryaev, Probability, in Graduate Texts in Math., 2nd ed. (Springer-Verlag, New York, 1996), Vol. 95.
F. Solomon, “Random Walks in a Random Environment”, Ann. Probab. 3, 1–31 (1975).
A. W. van der Vaart, Asymptotic Statistics, in Cambridge Series in Statist. and Probab. Math. (Cambridge Univ. Press, Cambridge, 1998), Vol. 3.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Falconnet, M., Gloter, A. & Loukianova, D. Maximum likelihood estimation in the context of a sub-ballistic random walk in a parametric random environment. Math. Meth. Stat. 23, 159–175 (2014). https://doi.org/10.3103/S1066530714030016
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066530714030016
Keywords
- asymptotic normality
- Cramér-Rao efficiency
- maximum likelihood estimation
- random walk in random environment