The cumulative distribution or probability density of a random variable, which is itself a function of a high number of independent real-valued random variables, can be formulated as high-dimensional integrals of an indicator or a Dirac $\delta$ function, respectively. To approximate the distribution or density at a point, we carry out preintegration with respect to one suitably chosen variable, then apply a Quasi-Monte Carlo method to compute the integral of the resulting smoother function. Interpolation is then used to reconstruct the distribution or density on an interval. We provide rigorous regularity and error analysis for the preintegrated function to show that our estimators achieve nearly first order convergence. Numerical results support the theory.

翻译：随机变量的累积分布或概率密度本身是大量独立实际估价随机变量的函数,可以分别作为指标或Dirac$\delta$函数的高维元件或高维元件。为了在某个点上大致显示分布或密度,我们先对一个适当选择的变量进行预整合,然后应用 Quasi-Monte Carlo 方法来计算由此产生的平滑函数的有机体。然后使用 Indigation 来在间隔内重建分布或密度。我们为预集函数提供严格的规律和错误分析,以显示我们的估计数几乎达到第一种顺序趋同。数字结果支持这一理论。