In this paper, we analyse a method for approximating the distribution function and density of a random variable that depends in a non-trivial way on a possibly high number of independent random variables, each with support on the whole real line. Starting with the integral formulations of the distribution and density, the method involves smoothing the original integrand by preintegration with respect to one suitably chosen variable, and then applying a suitable quasi-Monte Carlo (QMC) method to compute the integral of the resulting smoother function. Interpolation is then used to reconstruct the distribution or density on an interval. The preintegration technique is a special case of conditional sampling, a method that has previously been applied to a wide range of problems in statistics and computational finance. In particular, the pointwise approximation studied in this work is a specific case of the conditional density estimator previously considered in L'Ecuyer et al., arXiv:1906.04607. Our theory provides a rigorous regularity analysis of the preintegrated function, which is then used to show that the errors of the pointwise and interpolated estimators can both achieve nearly first-order convergence. Numerical results support the theory.

翻译：在本文中, 我们分析一种方法, 接近一个随机变量的分布函数和密度, 该随机变量的分布函数和密度以非三重方式取决于数量可能很多的独立随机变量, 每个变量都有整个真实线上的支持。 从分布和密度的综合配方开始, 该方法涉及以一个适当选择的变量来平滑原始整形, 然后应用一个合适的准蒙特卡洛( QMC) 方法来计算由此产生的平滑函数的整体。 然后, 内插用于重建一个间隔线上的分布或密度。 预一体化技术是有条件抽样的一个特殊案例, 这种方法以前曾应用于统计和计算融资方面的一系列广泛问题。 特别是, 这项工作研究的点近似近似性是先前在L' Ecuyer 等人, arXiv:1906.04607. 我们的理论提供了一种精确的组合前函数的规律性分析, 然后用来显示点和间断的估量器的误差可以达到第一级理论。