Title: Estimating accuracy of the MCMC variance estimator: a central limit theorem for batch means estimators Show Chinese title

Title: 估计MCMC方差估计器的准确性：批次均值估计器的中心极限定理

Abstract: The batch means estimator of the MCMC variance is a simple and effective measure of accuracy for MCMC based ergodic averages. Under various regularity conditions, the estimator has been shown to be consistent for the true variance. However, the estimator can be unstable in practice as it depends directly on the raw MCMC output. A measure of accuracy of the batch means estimator itself, ideally in the form of a confidence interval, is therefore desirable. The asymptotic variance of the batch means estimator is known; however, without any knowledge of asymptotic distribution, asymptotic variances are in general insufficient to describe variability. In this article we prove a central limit theorem for the batch means estimator that allows for the construction of asymptotically accurate confidence intervals for the batch means estimator. Additionally, our results provide a Markov chain analogue of the classical CLT for the sample variance parameter for i.i.d. observations. Our result assumes standard regularity conditions similar to the ones assumed in the literature for proving consistency. Simulated and real data examples are included as illustrations and applications of the CLT. Show Chinese abstract

Abstract: MCMC 方差的批量均值估计器是基于 MCMC 的遍历平均值的一种简单而有效的精度度量。在各种正则性条件下，该估计器已被证明是一致的，即收敛到真实方差。然而，在实际应用中，由于其直接依赖原始的 MCMC 输出，因此可能会不稳定。因此，理想情况下，批量均值估计器本身的精度度量（最好是置信区间的形式）是有价值的。 批量均值估计器的渐近方差是已知的；但是，如果没有关于渐近分布的知识，那么渐近方差通常不足以描述变异性。本文我们证明了批量均值估计器的一个中心极限定理，该定理允许构造批量均值估计器的渐近精确置信区间。此外，我们的结果提供了马尔可夫链版本的经典样本方差参数的中心极限定理，适用于独立同分布 (i.i.d.) 观测。 我们的结果假设的标准正则性条件与文献中用于证明一致性的条件类似。文中还包括模拟和真实数据的例子，作为中心极限定理的说明和应用。