Unbiased time-average estimators for Markov chains

TL;DR

This paper introduces an unbiased estimator for the long-term average of a functional of a Markov chain, improving estimation accuracy and efficiency under certain conditions.

Contribution

It presents a novel unbiased estimator for Markov chain functionals that is square-integrable, finite in expected running time, and asymptotically as efficient as traditional estimators.

Findings

The unbiased estimator converges to the true mean under coupling assumptions.

It is computationally feasible without precomputations in certain cases.

Numerical experiments confirm theoretical efficiency and unbiasedness.

Abstract

We consider a time-average estimator $f_{k}$ of a functional of a Markov chain. Under a coupling assumption, we show that the expectation of $f_{k}$ has a limit $\mu$ as the number of time-steps goes to infinity. We describe a modification of $f_{k}$ that yields an unbiased estimator $\hat f_{k}$ of $\mu$. It is shown that $\hat f_{k}$ is square-integrable and has finite expected running time. Under certain conditions, $\hat f_{k}$ can be built without any precomputations, and is asymptotically at least as efficient as $f_{k}$, up to a multiplicative constant arbitrarily close to $1$. Our approach provides an unbiased estimator for the bias of $f_{k}$. We study applications to volatility forecasting, queues, and the simulation of high-dimensional Gaussian vectors. Our numerical experiments are consistent with our theoretical findings.