On Berry-Esseen Rates for Jackknife Estimators

Abstract

Consider an ordinary estimation problem for an unknown parameter $\theta$. Let the estimator $\theta^\ast_n$ be the jackknife of a function of a $U$-statistic. Under mild assumptions, we demonstrate that $\sup_t |P\lbrack n^{1/2}(\theta^\ast_n - \theta)/S^\ast_n \leq t \rbrack - \Phi (t)| = O(n^{-p/2(p+1)})$, where $S^{\ast 2}_n$ is a jackknife estimator of the asymptotic variance of $n^{1/2}\theta^\ast_n, \Phi (t)$ is the standard normal distribution and $p$ is some positive number.