Affine matrix scrambling achieves smoothness-dependent convergence rates

Abstract

We study the convergence rate of the median estimator for affine matrix scrambled digital nets applied to integrands over the unit hypercube $[0, 1]^s$. By taking the median of $(2r-1)$ independent randomized quasi-Monte Carlo (RQMC) samples, we demonstrate that the desired convergence rates can be achieved without increasing the number of randomizations $r$ as the quadrature size $N$ grows for both bounded and unbounded integrands. For unbounded integrands, our analysis assumes a boundary growth condition on the weak derivatives and also considers singularities such as kinks and jump discontinuities. Notably, when $r = 1$, the median estimator reduces to the standard RQMC estimator. By applying analytical techniques developed for median estimators, we prove that the affine matrix scrambled estimator achieves a convergence rate depending on the integrand's smoothness, and is therefore not limited by the canonical rate $\mathcal{O}(N^{-3/2})$. However, this smoothness-dependent theoretical rate is not observed empirically in numerical experiments when the affine matrix scrambling yields a heavy-tailed sampling distribution. In contrast, the median estimator consistently reveals the theoretical rates and yields smaller integration errors than mean estimators, further highlighting its advantages.