On quantile estimation and Markov chain Monte Carlo convergence

In this paper we examine the method of Raftery & Lewis (1992) for estimating the convergence of Markov chain Monte Carlo, samplers when functionals of interest are in the form of parameter quantiles. Although popular, the method is commonly mis-applied to problems where quantiles are not of primary interest. We show how the method can be misleading in this case, and that it can seriously underestimate the true length of the burn-in. We provide a number of examples, comparing the convergence rate of the chain in respect of a particular quantile with that of the true convergence rate of the original chain. In particular we show how, in the case of the independence sampler, the two convergence rates are identical if the quantile of interest is chosen to be at an extreme of an appropriately reordered state space.