Mean squared error reduction of (plug-in) look-ahead estimators for Markov chains
Abstract
We consider the estimation of the invariant distribution π of a Markov kernel on a discrete state space (or of the mean value π(f) of a functional f under π) when solving the invariance equation is not feasible. Look-ahead estimators exploit the knowledge of the kernel by propagating the empirical occupation measure through k steps of the chain. When the kernel is known, as in Markov chain Monte Carlo, we compare their mean squared error with that of the empirical estimator, for independent or Markovian data, stationary or not. Our main result establishes that, for Markovian data, the one-step estimator always improves on the empirical one asymptotically, and that the improvement grows with k for reversible kernels. A non-reversible counterexample shows that reversibility cannot simply be dropped. In the statistical setting where the kernel belongs to a parametric family with unknown parameter, we introduce plug-in look-ahead estimators, which perform the look-ahead step with an estimated kernel. When the parameter is estimated at the parametric rate, we give a necessary and sufficient condition under which the plug-in look-ahead estimator outperforms the empirical one. The theoretical results are illustrated by extensive simulation studies, which also shed further light on the behaviour of the estimators at hand. Both settings are finally applied to real data arising from geographical mobility problems in the Lyon metropolitan area, France.
Related
- Matrix asymptotic calculus for plug-in maximum likelihood estimators in finite Markov chains
- Phase transitions and approximations of mean squared error for state-space models with fractional differencing
- Bernstein’s inequalities for general Markov chains
- Partition of unity a posteriori error estimators for the linear elasticity eigenvalue problem
- Global central limit theorems for Markov chains
BibTeX
@misc{azais2026mean,
title = {{Mean squared error reduction of (plug-in) look-ahead estimators for Markov chains}},
author = {Romain Azaïs and Benoît Henry and Henri Péchoux},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04069},
eprint = {2610.04069},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04069},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mean squared error reduction of (plug-in) look-ahead estimators for Markov chains
AU - Romain Azaïs
AU - Benoît Henry
AU - Henri Péchoux
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.04069
UR - https://arxiv.org/abs/2610.04069
AB - We consider the estimation of the invariant distribution π of a Markov kernel on a discrete state space (or of the mean value π(f) of a functional f under π) when solving the invariance equation is not feasible. Look-ahead estimators exploit the knowledge of the kernel by propagating the empirical occupation measure through k steps of the chain. When the kernel is known, as in Markov chain Monte Carlo, we compare their mean squared error with that of the empirical estimator, for independent or Markovian data, stationary or not. Our main result establishes that, for Markovian data, the one-step estimator always improves on the empirical one asymptotically, and that the improvement grows with k for reversible kernels. A non-reversible counterexample shows that reversibility cannot simply be dropped. In the statistical setting where the kernel belongs to a parametric family with unknown parameter, we introduce plug-in look-ahead estimators, which perform the look-ahead step with an estimated kernel. When the parameter is estimated at the parametric rate, we give a necessary and sufficient condition under which the plug-in look-ahead estimator outperforms the empirical one. The theoretical results are illustrated by extensive simulation studies, which also shed further light on the behaviour of the estimators at hand. Both settings are finally applied to real data arising from geographical mobility problems in the Lyon metropolitan area, France.
ER -