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Unbiased time-average estimators for Markov chains

Nabil Kahale

TL;DR

The authors address bias in time-average estimators for Markov chains by constructing an unbiased estimator hat f_k through a bias-correction mechanism rooted in a randomized multilevel framework. They establish convergence and variance bounds for conventional estimators under a coupling assumption, then derive unbiased estimators for the bias via a single-term estimator and a carefully chosen parameterization, including oblivious and stratified variants. The approach yields estimators that are square-integrable with finite running time and, under mild conditions, asymptotically match or surpass the efficiency of standard time averages, while enabling practical bias assessment and confidence interval construction. Numerical experiments on GARCH volatility, queuing systems, and high-dimensional Gaussian vectors corroborate the theory and demonstrate robust performance even for discontinuous functionals.

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 $μ$ 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 $μ$. 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.

Unbiased time-average estimators for Markov chains

TL;DR

The authors address bias in time-average estimators for Markov chains by constructing an unbiased estimator hat f_k through a bias-correction mechanism rooted in a randomized multilevel framework. They establish convergence and variance bounds for conventional estimators under a coupling assumption, then derive unbiased estimators for the bias via a single-term estimator and a carefully chosen parameterization, including oblivious and stratified variants. The approach yields estimators that are square-integrable with finite running time and, under mild conditions, asymptotically match or surpass the efficiency of standard time averages, while enabling practical bias assessment and confidence interval construction. Numerical experiments on GARCH volatility, queuing systems, and high-dimensional Gaussian vectors corroborate the theory and demonstrate robust performance even for discontinuous functionals.

Abstract

We consider a time-average estimator of a functional of a Markov chain. Under a coupling assumption, we show that the expectation of has a limit as the number of time-steps goes to infinity. We describe a modification of that yields an unbiased estimator of . It is shown that is square-integrable and has finite expected running time. Under certain conditions, can be built without any precomputations, and is asymptotically at least as efficient as , up to a multiplicative constant arbitrarily close to . Our approach provides an unbiased estimator for the bias of . We study applications to volatility forecasting, queues, and the simulation of high-dimensional Gaussian vectors. Our numerical experiments are consistent with our theoretical findings.
Paper Structure (36 sections, 26 theorems, 158 equations, 1 figure, 9 tables, 4 algorithms)

This paper contains 36 sections, 26 theorems, 158 equations, 1 figure, 9 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Conventional time-average estimators
  3. The coupling assumption
  4. Convergence properties
  5. Sharpness of bounds
  6. Unbiased time-average estimators
  7. The single term estimator
  8. Construction of $\hat{f}_{k}$
  9. $\nu$-dependent parameters
  10. Oblivious parameters
  11. A stratified unbiased estimator
  12. Implementation details
  13. Examples
  14. GARCH volatility model
  15. $GI/G/1$ queue
  16. ...and 21 more sections

Key Result

Proposition 2.1

Suppose there is $x\in F$ and a positive decreasing sequence $(\nu'(i),i\geq0)$ that satisfies eq:infiniteSumAssumption and, for $i,m\geq0$, Then Assumption A1 holds with $\nu(i)=4\nu'(i)$ for $i\geq0$.

Figures (1)

  • Figure 1: Absolute bias and standard deviation of time-average estimators with $10^{6}$ independent replications and burn-in period $b(k)=\lfloor k/10\rfloor$.

Theorems & Definitions (40)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Theorem 2.1
  • Lemma 2.1
  • Proposition 2.3
  • Theorem 3.1: GlynnRhee2015unbiased
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 30 more