Quantiles and expected shortfalls are usually used to measure risks of stochastic systems, which are often estimated by Monte Carlo methods. This paper focuses on the use of quasi-Monte Carlo (QMC) method, whose convergence rate is asymptotically better than Monte Carlo in the numerical integration. We first prove the convergence of QMC-based quantile estimates under very mild conditions, and then establish a deterministic error bound of $O(N^{-1/d})$ for the quantile estimates, where $d$ is the dimension of the QMC point sets used in the simulation and $N$ is the sample size. Under certain conditions, we show that the mean squared error (MSE) of the randomized QMC estimate for expected shortfall is $o(N^{-1})$. Moreover, under stronger conditions the MSE can be improved to $O(N^{-1-1/(2d-1)+\epsilon})$ for arbitrarily small $\epsilon>0$.

中文翻译：

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准蒙特卡罗采样的分位数和预期短缺的收敛分析

分位数和预期缺口通常用于衡量随机系统的风险，通常通过蒙特卡罗方法进行估计。本文重点介绍了拟蒙特卡罗(QMC)方法的使用，其收敛速度在数值积分上渐近优于蒙特卡罗。我们首先证明了基于 QMC 的分位数估计在非常温和的条件下的收敛性，然后为分位数估计建立了 $O(N^{-1/d})$ 的确定性误差界限，其中 $d$ 是模拟中使用的 QMC 点集，$N$ 是样本大小。在某些条件下，我们表明预期短缺的随机 QMC 估计的均方误差 (MSE) 是 $o(N^{-1})$。此外，在更强的条件下，对于任意小的 $\epsilon>0$，MSE 可以改进为 $O(N^{-1-1/(2d-1)+\epsilon})$。