Negative Dependence, Scrambled Nets, and Variance Bounds
本文建立了一个研究随机化拟蒙特卡洛方法抽样方案相依结构的框架，推广了Hoeffding引理，并证明了打乱(0,m,s)-网等负相依抽样方案能比蒙特卡洛方法获得更小的估计方差。
In this paper, we provide a framework to study the dependence structure of sampling schemes such as those produced by randomized quasi-Monte Carlo methods. The main goal of this new framework is to determine conditions under which the negative dependence structure of a sampling scheme enables the construction of estimators with reduced variance compared to Monte Carlo estimators. To do this, we establish a generalization of the well-known Hoeffding’s lemma—expressing the covariance of two random variables as an integral of the difference between their joint distribution function and the product of their marginal distribution functions—that is particularly well suited to study such sampling schemes. We also provide explicit formulas for the joint distribution of pairs of points randomly chosen from a scrambled (0, m, s)-net. In addition, we provide variance bounds establishing the superiority of dependent sampling schemes over Monte Carlo in a few different setups. In particular, we show that a scrambled (0, m, 2)-net yields an estimator with variance no larger than a Monte Carlo estimator for functions monotone in each variable.