Skewness of a randomized quasi-Monte Carlo estimate

TL;DR

This paper investigates the skewness properties of randomized quasi-Monte Carlo estimates, providing theoretical bounds and probabilistic analyses that explain their observed reliability in confidence interval estimation.

Contribution

It establishes conditions under which the skewness of RQMC estimates remains small and improves probabilistic bounds on the distribution of the quality parameter for digital nets.

Findings

Skewness of RQMC estimates can be bounded by $O(n^psilon)$

Under a random generator matrix model, skewness improves to $O(n^{-1/2+psilon})$

Probabilistic bounds on the distribution of the quality parameter are enhanced

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 $O(n^\epsilon)$ for any $\epsilon>0$, so 'almost $O(1)$'. Under a random generator matrix model, we can improve this rate to $O(n^{-1/2+\epsilon})$ 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.