Asymptotic distributions of a new type of design‐based incomplete U‐statistics

Abstract

The U‐statistic has been an important part of the arsenal of statistical tools. Meanwhile, the computation of it could easily become expensive. As a remedy, the idea of incomplete U‐statistics has been adopted in practice, where only a small fraction of combinations of units are evaluated. Recently, researchers proposed a new type of incomplete U‐statistics called ICUDO, which needs substantially less time of computing than all existing methods. This paper aims to study the asymptotic distributions of ICUDO to facilitate the corresponding statistical inference. This is a non‐trivial task due to the restricted randomization in the sampling scheme of ICUDO. The bootstrap approach for the finite sample distribution of ICUDO is also discussed. Lastly, we observe some intrinsic connections between U‐statistics and computer experiments in the context of integration approximation. This allows us to generalize some existing theoretical results in the latter topic.