Negative Dependence, Scrambled Nets, and Variance Bounds

• 成果类型: Article
• 署名作者: Lemieux, Christiane
• 署名单位: University of Waterloo
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2017.0861
• 发表日期: 2018
• 页码: 228-251
• 关键词: monte-carlo; integration

摘要

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.