We introduce a class of Monte Carlo estimators that aim to overcome the rapid growth of variance with dimension often observed for standard estimators by exploiting the target's independence structure. We identify the most basic incarnations of these estimators with a class of generalized U-statistics, and thus establish their unbiasedness, consistency, and asymptotic normality. Moreover, we show that they obtain the minimum possible variance amongst a broad class of estimators; and we investigate their computational cost and delineate the settings in which they are most efficient. We exemplify the merger of these estimators with other well-known Monte Carlo estimators so as to better adapt the latter to the target's independence structure and improve their performance. We do this via three simple mergers: one with importance sampling, another with importance sampling squared, and a final one with pseudo-marginal Metropolis-Hasting. In all cases, we show that the resulting estimators are well-founded and achieve lower variances than their standard counterparts. Lastly, we illustrate the various variance reductions through several examples.
Inference for Multi-Dimensional High-Frequency Data: Equivalence of Methods, Central Limit Theorems, and an Application to Conditional Independence Testing
The Minimax Rate of HSIC Estimation for Translation-Invariant Kernels
Institute for Program Structures and Data Organization（程序结构与数据组织研究所）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；Department of Statistics（统计系）；London School of Economics（伦敦政治经济学院）
