LOCAL ANTITHETIC SAMPLING WITH SCRAMBLED NETS

• 成果类型: Article
• 署名作者: Owen, Art B.
• 署名单位: Stanford University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/07-AOS548
• 发表日期: 2008
• 页码: 2319-2343
• 关键词: monomial cubature rules; digital nets; monte-carlo; INTEGRALS; variance; compilation; discrepancy; SEQUENCES; stroud

摘要

We consider the problem of computing an approximation to the integral I = integral([0, 1])d f (x) dx. Monte Carlo (MC) sampling typically attains a root mean squared error (RMSE) of O(n(-1/2)) from n independent random function evaluations. By contrast, quasi-Monte Carlo (QMC) sampling using carefully equispaced evaluation points can attain the rate O(n(-1+epsilon)) for any epsilon > 0 and randomized QMC (RQMC) can attain the RMSE O(n(-3/2+epsilon)), both under mild conditions on f. Classical variance reduction methods for MC call be adapted to QMC. Published results combining QMC with importance sampling and with control variates have found worthwhile improvements, but no change in the error rate. This paper extends the classical variance reduction method of antithetic sampling and combines it with RQMC. One Such method is shown to bring a modest improvement in the RMSE rate, attaining O(n(-3/2-1/d+epsilon)) for any epsilon > 0, for smooth enough f.