标题： 带有分类变量的准蒙特卡罗方法 显示英文标题

标题： Quasi-Monte Carlo with one categorical variable

摘要： 我们研究了当其中一个变量仅取有限个值时的多变量积分的随机准蒙特卡罗（RQMC）估计。 这个问题出现在积分变量从混合分布中抽取的情况下，这在重要性抽样中很常见，也出现在一些关于传输映射的最新工作中。 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比使用比例分配更有利。 最优分配取决于可能未知的收敛速率。 在假设错误的速率下设计样本仍能达到该收敛速率，但会得到一个较差的隐含常数。 使用保守的速率所带来的惩罚通常比使用乐观的速率更高。 我们还发现，对于最精确的RQMC抽样方法，将$n=2^m$随机Sobol'点安排成也是$2$幂次的子样本大小是有利的。 显示英文摘要

摘要： 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 beneficial to oversample the smallest mixture components instead of using a proportional allocation. The optimal allocations depend on the possibly unknown convergence rates. 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 $2$.