We consider quantile estimation using Markov chain Monte Carlo and establish conditions under which the sampling distribution of the Monte Carlo error is approximately Normal. Further, we investigate techniques to estimate the associated asymptotic variance, which enables construction of an asymptotically valid interval estimator. Finally, we explore the finite sample properties of these methods through examples and provide some recommendations to practitioners.
"Markov chain Monte Carlo estimation of quantiles." Electron. J. Statist. 8 (2) 2448 - 2478, 2014. https://doi.org/10.1214/14-EJS957