The Berry‐Esseen bound for Studentized U ‐statistics

Callaert and Veraverbeke (1981) recently obtained a Berry‐Esseen‐type bound of order n –1/2 for Studentized nondegenerate U ‐statistics of degree two. The condition these authors need to obtain this order bound is the finiteness of the 4.5th absolute moment of the kernel h . In this note it is shown that this assumption can be weakened to that of a finite (4 + ϵ)th absolute moment of the kernel h , for some ϵ > 0. Our proof resembles part of Helmers and van Zwet (1982), where an analogous result is obtained for the Student t ‐statistic. The present note extends this to Studentized U ‐statistics.