Integration of bounded monotone functions: Revisiting the nonsequential case, with a focus on unbiased Monte Carlo (randomized) methods

TL;DR

This paper analyzes the numerical integration of monotone bounded functions using nonsequential Monte Carlo methods, providing new bounds on error and evaluating unbiased techniques like control variates and stratified sampling.

Contribution

It introduces improved lower bounds on the maximal error for nonsequential algorithms and assesses the performance of specific unbiased Monte Carlo methods.

Findings

New lower bound on $L^p$ error for nonsequential algorithms when p > 1.

Analysis of the maximal error for control variate and stratified sampling methods at p = 2.

Insights into the effectiveness of unbiased Monte Carlo techniques for monotone functions.

Abstract

In this article we revisit the problem of numerical integration for monotone bounded functions, with a focus on the class of nonsequential Monte Carlo methods. We first provide new a lower bound on the maximal $L^p$ error of nonsequential algorithms, improving upon a theorem of Novak when p > 1. Then we concentrate on the case p = 2 and study the maximal error of two unbiased methods-namely, a method based on the control variate technique, and the stratified sampling method.