Efficient shape-constrained inference for the autocovariance sequence from a reversible Markov chain

TL;DR

This paper introduces a new shape-constrained estimator for the autocovariance sequence of reversible Markov chains, providing strong theoretical guarantees and demonstrating superior empirical performance over existing methods.

Contribution

The paper proposes a novel shape-constrained estimator for autocovariance sequences that ensures consistency and improves variance estimation in Markov chain analysis.

Findings

The estimator is strongly consistent for autocovariance sequences.

It provides consistent estimates of the asymptotic variance.

Empirical results show improved accuracy over existing methods.

Abstract

In this paper, we study the problem of estimating the autocovariance sequence resulting from a reversible Markov chain. A motivating application for studying this problem is the estimation of the asymptotic variance in central limit theorems for Markov chains. We propose a novel shape-constrained estimator of the autocovariance sequence, which is based on the key observation that the representability of the autocovariance sequence as a moment sequence imposes certain shape constraints. We examine the theoretical properties of the proposed estimator and provide strong consistency guarantees for our estimator. In particular, for geometrically ergodic reversible Markov chains, we show that our estimator is strongly consistent for the true autocovariance sequence with respect to an $\ell_2$ distance, and that our estimator leads to strongly consistent estimates of the asymptotic variance. Finally, we perform empirical studies to illustrate the theoretical properties of the proposed estimator as well as to demonstrate the effectiveness of our estimator in comparison with other current state-of-the-art methods for Markov chain Monte Carlo variance estimation, including batch means, spectral variance estimators, and the initial convex sequence estimator.