Stochastic comparisons of stratified sampling techniques for some Monte Carlo estimators

TL;DR

This paper compares various stratified sampling estimators for the supremum and integral of a function, emphasizing stochastic order criteria to determine the effectiveness of different stratification levels.

Contribution

It introduces a stochastic order-based framework for comparing stratified sampling estimators, highlighting the benefits of refined stratification.

Findings

More refined stratification improves estimator performance.

Stochastic order comparisons provide stronger evaluation criteria.

Refined stratification is advantageous for supremum and integral estimation.

Abstract

We compare estimators of the (essential) supremum and the integral of a function $f$ defined on a measurable space when $f$ may be observed at a sample of points in its domain, possibly with error. The estimators compared vary in their levels of stratification of the domain, with the result that more refined stratification is better with respect to different criteria. The emphasis is on criteria related to stochastic orders. For example, rather than compare estimators of the integral of $f$ by their variances (for unbiased estimators), or mean square error, we attempt the stronger comparison of convex order when possible. For the supremum, the criterion is based on the stochastic order of estimators.