First- and second-order error estimates in Monte Carlo integration

TL;DR

This paper discusses the importance of accurate error estimation in Monte Carlo integration, introducing a second-order unbiased estimator and addressing its computational challenges and slow convergence.

Contribution

It presents a novel second-order unbiased error estimator for Monte Carlo integration and proposes an efficient linear-time computation method.

Findings

The second-order estimator can be unbiased but not necessarily positive.

A linear-time computation method for the second-order estimator is proposed.

The second-order error estimate converges slowly in practice.

Abstract

In Monte Carlo integration an accurate and reliable determination of the numerical intregration error is essential. We point out the need for an independent estimate of the error on this error, for which we present an unbiased estimator. In contrast to the usual (first-order) error estimator, this second-order estimator can be shown to be not necessarily positive in an actual Monte Carlo computation. We propose an alternative and indicate how this can be computed in linear time without risk of large rounding errors. In addition, we comment on the relatively very slow convergence of the second-order error estimate.