The median trick does not help for fully nested scrambling

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.