Geweke (1992) proposed a convergence diagnostic for Markov chains based on a test for equality of the means of the first and last part of a Markov chain (by default the first 10% and the last 50%). If the samples are drawn from the stationary distribution of the chain, the two means are equal and Geweke's statistic has an asymptotically standard normal distribution.
The test statistic is a standard Z-score: the difference between the two sample means divided by its estimated standard error. The standard error is estimated from the spectral density at zero and so takes into account any autocorrelation.
The Z-score is calculated under the assumption that the two parts of
the chain are asymptotically independent, which requires that the sum
of frac1 and frac2 be strictly less than 1.
geweke.diag(x, frac1=0.1, frac2=0.5)an mcmc object
fraction to use from beginning of chain
fraction to use from end of chain
Z-scores for a test of equality of means between the first and last parts of the chain. A separate statistic is calculated for each variable in each chain.
Geweke, J. Evaluating the accuracy of sampling-based approaches to calculating posterior moments. In Bayesian Statistics 4 (ed JM Bernado, JO Berger, AP Dawid and AFM Smith). Clarendon Press, Oxford, UK.