Robust estimation for ergodic Markovian processes

TL;DR

This paper introduces a robust method for estimating the stationary distribution of ergodic Markov processes from dependent data, with guarantees on performance even under misspecification and contamination.

Contribution

It provides a non-asymptotic deviation bound for the estimator and demonstrates robustness and near-i.i.d. performance under various dependence conditions.

Findings

Estimator achieves near-i.i.d. performance with weak dependence.

Method is robust to model misspecification and contamination.

Applicable to diffusion processes and hidden Markov models.

Abstract

We observe n possibly dependent random variables, the distribution of which is presumed to be stationary even though this might not be true, and we aim at estimating the stationary distribution. We establish a non-asymptotic deviation bound for the Hellinger distance between the target distribution and our estimator. If the dependence within the observations is small, the estimator performs as good as if the data were independent and identically distributed. In addition our estimator is robust to misspecification and contamination. If the dependence is too high but the observed process is mixing, we can select a subset of observations that is almost independent and retrieve results similar to what we have in the i.i.d. case. We apply our procedure to the estimation of the invariant distribution of a diffusion process and to finite state space hidden Markov models.