The median trick does not help for fully nested scrambling

TL;DR

This paper demonstrates that the median trick does not improve convergence in fully nested scrambled digital nets for numerical integration, unlike in linear scrambling.

Contribution

It shows that median estimators with fully nested scrambling do not gain the convergence benefits seen with linear scrambling in quasi-Monte Carlo methods.

Findings

Median estimator with linear scrambling achieves faster convergence for smooth functions.

Median estimator with fully nested scrambling does not improve convergence.

Equivalence in variance between average estimators does not extend to median estimators.

Abstract

In randomized quasi-Monte Carlo methods for numerical integration, average estimators based on digital nets with fully nested and linear scrambling are known to exhibit the same variance. In this note, we show that this equivalence does not extend to the median estimators. Specifically, while the median estimator with linear scrambling can achieve faster convergence for smooth integrands, the median estimator with fully nested scrambling does not exhibit this advantage.