  Estimating Monte Carlo error is critical to valid simulation results in
Markov chain Monte Carlo (MCMC) and initial sequence estimators were one of the
first methods introduced for this. Over the last few years, focus has been on
multivariate assessment of simulation error, and many multivariate
generalizations of univariate methods have been developed. The multivariate
initial sequence estimator is known to exhibit superior finite-sample
performance compared to its competitors. However, the multivariate initial
sequence estimator can be prohibitively slow, limiting its widespread use. We
provide an efficient alternative to the multivariate initial sequence estimator
that inherits both its asymptotic properties as well as the finite-sample
superior performance. The effectiveness of the proposed estimator is shown via
some MCMC example implementations. Further, we also present univariate and
multivariate initial sequence estimators for when parallel MCMC chains are run
and demonstrate their effectiveness over popular alternative.
