Analysis of preintegration followed by quasi-Monte Carlo integration for distribution functions and densities

TL;DR

This paper investigates a combined preintegration and quasi-Monte Carlo approach for accurately approximating distribution functions and densities of complex, high-dimensional random variables, with theoretical error analysis and numerical validation.

Contribution

It provides a rigorous analysis of the regularity and convergence properties of the preintegrated quasi-Monte Carlo method for distribution and density estimation.

Findings

Achieves nearly first-order convergence in approximation errors.

Provides a regularity analysis of the preintegrated functions.

Numerical results support the theoretical convergence rates.

Abstract

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.