SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 1, Page 280-301, January 2021.
We consider the problem of estimating the density of a random variable $X$ that can be sampled exactly by Monte Carlo (MC). We investigate the effectiveness of replacing MC by randomized quasi-MC (RQMC) or by stratified sampling over the unit cube to reduce the integrated variance (IV) and the mean integrated square error (MISE) for kernel density estimators. We show theoretically and empirically that the RQMC and stratified estimators can achieve substantial reductions of the IV and the MISE, and even faster convergence rates than MC in some situations, while leaving the bias unchanged. We also show that the variance bounds obtained via a traditional Koksma--Hlawka-type inequality for RQMC are much too loose to be useful when the dimension of the problem exceeds a few units. We describe an alternative way to estimate the IV, a good bandwidth, and the MISE, under RQMC or stratification, and we show empirically that in some situations, the MISE can be reduced significantly even in high-dimensional settings.

中文翻译：

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随机拟蒙特卡罗算法的密度估计

SIAM / ASA不确定性量化杂志，第9卷，第1期，第280-301页，2021年1月。
我们考虑估计可以由Monte Carlo（MC）精确采样的随机变量$ X $的密度的问题。我们研究了通过随机准MC（RQMC）或通过对单位立方进行分层抽样来减少MC的有效性，以减少积分密度（IV）和平均积分平方误差（MISE）的核密度估计量。我们从理论和经验上证明，在某些情况下，RQMC和分层估计量可以实现IV和MISE的大幅降低，甚至比MC更快的收敛速度，同时保持偏差不变。我们还表明，通过RQMC的传统Koksma-Hlawka型不等式获得的方差边界过于宽松，以至于当问题的维数超过几个单位时就无法使用。我们描述了另一种估算IV，良好带宽的方法，