Skewness of a randomized quasi-Monte Carlo estimate

TL;DR

The paper analyzes the skewness of randomized quasi-Monte Carlo estimates obtained from scrambled digital nets and links this property to the reliability of $t$-based confidence intervals. By employing a Walsh-Fourier decomposition and careful treatment of symmetric error contributions, it proves that the third central moment decays as $O(n^{-9/2+\varepsilon})$ in general and as $O(n^{-5+\varepsilon})$ under random generator matrices, yielding a skewness of $\gamma=O(n^{-1/2+\varepsilon})$ when variance scales as $\Omega(n^{-3})$. It also provides improved probabilistic bounds for the distribution of the net quality parameter $t$ under random base-$p$ generators and discusses refinements for the case of random generator matrices, which hold with high probability. Through explicit 1D and 3D examples, the results align with observed symmetry and validate the theoretical decay of skewness, offering justification for the effectiveness of standard $t$-intervals in ci4rqmc while highlighting limitations for non-smooth integrands and fixed nets.

Abstract

Some recent work on confidence intervals for randomized quasi-Monte Carlo (RQMC) sampling found a surprising result: ordinary Student $t$ 95% confidence intervals based on a modest number of replicates were seen to be very effective and even more reliable than some bootstrap $t$ intervals that were expected to be best. One potential explanation is that those RQMC estimates have small skewness. In this paper we give conditions under which the skewness is $O(n^ε)$ for any $ε>0$, so 'almost $O(1)$'. Under a random generator matrix model, we can improve this rate to $O(n^{-1/2+ε})$ with very high probability. We also improve some probabilistic bounds on the distribution of the quality parameter $t$ for a digital net in a prime base under random sampling of generator matrices.

Figures (1)

• Figure 1: This is a sketch of the sets $\kappa_1$, $\kappa_2$, $\kappa_{12}$, when all are non-empty. Solid circles mark indices that must be in those sets including $\ell_1=\lceil\kappa_1\rceil$ and $\ell_2=\lceil\kappa_2\rceil=\lceil\kappa_{12}\rceil$. Without loss of generality $\ell_1\in\kappa_{12}\setminus\kappa_2$. Open circles in $\kappa_1$ and $\kappa_2$ indicate possible locations of $\ell_3=\lceil\kappa_1\rceil_{(2)}$ and $\ell_4=\lceil\kappa_2\rceil_{(2)}$, respectively. Every index must be in exactly $0$ or $2$ of sets $\kappa_1$, $\kappa_2$, $\kappa_{12}$.

Theorems & Definitions (24)

• Theorem 1
• proof
• Theorem 2
• proof
• Proposition 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof