On the second order efficiency of bootstrap estimators of sampling distributions (Q1113237)

Given \(X_ i\) i.i.d. with d.f. F, let \(g_ n(.,F)\) be the d.f. of an appropriate normalized statistic based on \(X_ 1,...,X_ n\) and let \(\hat g_{n,B}=g_ n(.,\hat F_ n)\) be the bootstrap estimator of \(g_ n(.,F)\), where \(\hat F_ n\) is the empirical d.f. based on \(X_ 1,...,X_ n\). The present paper proves second order asymptotic efficiency in the sense of \textit{M. Akahira} and \textit{K. Takeuchi} [Asymptotic efficiency of statistical estimators: concepts and higher order asymptotic efficiency. (1981; Zbl 0463.62026)], of an appropriately corrected version of \(\hat g_{n,B}\) under conditions about \(g_ n(.,F)\) similar to Ass. 1 or 1' of \textit{R. Beran} [Ann. Stat. 10, 212-225 (1982; Zbl 0485.62037)].