We study randomized quasi-Monte Carlo (RQMC) estimation of a multivariate integral where one of the variables takes only a finite number of values. This problem arises when the variable of integration is drawn from a mixture distribution as is common in importance sampling and also arises in some recent work on transport maps. We find that when integration error decreases at an RQMC rate that it is then important to oversample the smallest mixture components instead of using a proportional allocation. This can even improve the rate of convergence. The optimal allocations depend on the possibly unknown convergence rate. Designing the sample with an incorrect assumption on the rate still attains that convergence rate, with an inferior implied constant. The penalty for using a pessimistic rate is typically higher than for using an optimistic one. We also find that for the most accurate RQMC sampling methods, it is advantageous to arrange that our $n=2^m$ randomized Sobol' points split into subsample sizes that are also powers of $2$.

翻译：我们研究了当其中一个变量仅取有限个值时，多元积分的随机化拟蒙特卡洛（RQMC）估计问题。该问题出现在积分变量来自混合分布的情形中，这在重要性抽样中很常见，并且在近期关于传输映射的一些工作中也会出现。我们发现，当积分误差以RQMC速率下降时，对最小的混合成分进行过采样而非采用比例分配变得尤为重要。这甚至可以提高收敛速率。最优分配取决于可能未知的收敛速率。基于对速率的错误假设设计样本，仍能达到该收敛速率，但隐含常数会变差。使用悲观速率通常比使用乐观速率的惩罚更高。我们还发现，对于最精确的RQMC抽样方法，将我们的 $n=2^m$ 个随机化Sobol'点划分为同样为2的幂次的子样本量是有利的。