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.

翻译：在马尔可夫链蒙特卡洛（MCMC）中，估计蒙特卡洛误差对于模拟结果的有效性至关重要，而初始序列估计器是最早为此引入的方法之一。近年来，研究重点转向模拟误差的多元评估，并已发展出许多单变量方法的多元推广。已知多元初始序列估计器相比其竞争方法展现出更优的有限样本性能。然而，多元初始序列估计器的计算速度可能极其缓慢，这限制了其广泛应用。我们提出一种高效的多元初始序列估计器替代方案，该方案既继承了原估计器的渐近性质，又保持了其有限样本的优越性能。通过若干MCMC示例实现，展示了所提估计器的有效性。此外，我们还针对并行运行MCMC链的情形提出了单变量与多元初始序列估计器，并证明了它们相较于常用替代方法的优越性。