Abstract: Markov chain Monte Carlo (MCMC) simulations are commonly employed for estimating features of a target distribution, particularly for Bayesian inference. A fundamental challenge is determining when these simulations should stop. We consider a sequential stopping rule that terminates the simulation when the width of a confidence interval is sufficiently small relative to the size of the target parameter. Specifically, we propose relative magnitude and relative standard deviation stopping rules in the context of MCMC. In each setting, we develop conditions to ensure the simulation will terminate with probability one and the resulting confidence intervals will have the proper coverage probability. Our results are applicable in such MCMC estimation settings as expectation, quantile, or simultaneous multivariate estimation. We investigate the finite sample properties through a variety of examples, and provide some recommendations to practitioners.

Key words and phrases: Batch means, Bayesian computation, fixed-width confidence intervals, sequential estimation, sequential stopping rules, strong consistency.