Randomized quasi-Monte Carlo (RQMC) methods estimate the mean of a random variable by sampling an integrand at $n$ equidistributed points. For scrambled digital nets, the resulting variance is typically $\tilde O(n^{-θ})$ where $θ\in[1,3]$ depends on the smoothness of the integrand and $\tilde O$ neglects logarithmic factors. While RQMC can be far more accurate than plain Monte Carlo (MC) it remains difficult to get confidence intervals on RQMC estimates. We investigate some empirical Bernstein confidence intervals (EBCI) and hedged betting confidence intervals (HBCI), both from Waudby-Smith and Ramdas (2024), when the random variable of interest is subject to known bounds. When there are $N$ integrand evaluations partitioned into $R$ independent replicates of $n=N/R$ RQMC points, and the RQMC variance is $Θ(n^{-θ})$, then an oracle minimizing the width of a Bennett confidence interval would choose $n =Θ(N^{1/(θ+1)})$. The resulting intervals have a width that is $Θ(N^{-θ/(θ+1)})$. Our empirical investigations had optimal values of $n$ grow slowly with $N$, HBCI intervals that were usually narrower than the EBCI ones, and optimal values of $n$ for HBCI that were equal to or smaller than the ones for the oracle.

翻译：随机化拟蒙特卡洛（RQMC）方法通过在$n$个等分布点上对积分函数进行采样来估计随机变量的均值。对于加扰数字网，所得方差通常为$\tilde O(n^{-θ})$，其中$θ\in[1,3]$取决于积分函数的平滑度，而$\tilde O$忽略了对数因子。尽管RQMC可能比普通蒙特卡洛（MC）精确得多，但获得RQMC估计的置信区间仍然困难。我们研究了当目标随机变量受已知边界约束时，来自Waudby-Smith和Ramdas（2024）的经验伯恩斯坦置信区间（EBCI）和对冲投注置信区间（HBCI）。当$N$次积分函数评估被划分为$R$个独立的$n=N/R$点RQMC重复实验，且RQMC方差为$Θ(n^{-θ})$时，最小化Bennett置信区间宽度的预言机将选择$n =Θ(N^{1/(θ+1)})$。所得区间的宽度为$Θ(N^{-θ/(θ+1)})$。我们的实证研究表明：最优$n$值随$N$缓慢增长；HBCI区间通常比EBCI区间更窄；且HBCI的最优$n$值等于或小于预言机的最优值。