Skewness of a randomized quasi-Monte Carlo estimate

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.