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

TL;DR

The paper studies numerical integration of bounded monotone functions in a nonsequential Monte Carlo setting, reducing to $S(f)=\mathbb{E}[f(X)]=\int_0^1 f(x)\,dx$ with $X\sim U[0,1]$ and $f\in F$. It proves a general $L^p$ lower bound on the maximal error for nonsequential estimators, generalizing Novak's $L^1$ bound, and analyzes two unbiased $p=2$ strategies—control variates and stratified sampling—in terms of their worst-case variance. For uniform i.i.d. sampling, the maximal $L^2$ error is $e_2^{MC}=1/(2\sqrt{n})$, while the control variate reduces this to $e_2^{cv}=1/\sqrt{12n}$ and a staircase-approximation argument shows $\mathrm{var}(f(X)-X)\le 1/12$. Stratified sampling yields $e_2=1/(2n)$ in the optimal $K=n$, equal-strata case, with connections to Latin Hypercube Sampling in 1D. The results quantify variance/error trade-offs in one-dimensional monotone integration and guide design of rate-optimal unbiased nonsequential and, potentially, sequential methods, with open questions on optimality and extensions to higher dimensions.

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.

Theorems & Definitions (8)

• Remark 1
• Theorem 2.1
• Corollary 2.2
• proof : Proof of Theorem \ref{['th:lp']}
• Theorem 3.1
• proof
• Theorem 4.1
• proof