High‐order Corrected Estimator of Asymptotic Variance with Optimal Bandwidth
针对依赖过程样本均值的渐近方差，提出一类高阶校正估计量，在最优收敛速度下通过校正渐近偏差提升均方误差性能，适用于频率学派置信区间和贝叶斯有效样本量评估。
Abstract Estimation of time‐average variance constant (TAVC), which is the asymptotic variance of the sample mean of a dependent process, is of fundamental importance in various fields of statistics. For frequentists, it is crucial for constructing confidence interval of mean and serving as a normalizing constant in various test statistics and so forth. For Bayesians, it is widely used for evaluating effective sample size and conducting convergence diagnosis in Markov chain Monte Carlo method. In this paper, by considering high‐order corrections to the asymptotic biases, we develop a new class of TAVC estimators that enjoys optimal ‐convergence rates under different degrees of the serial dependence of stochastic processes. The high‐order correction procedure is applicable to estimation of the so‐called smoothness parameter, which is essential in determining the optimal bandwidth. Comparisons with existing TAVC estimators are comprehensively investigated. In particular, the proposed optimal high‐order corrected estimator has the best performance in terms of mean squared error.