Jackknife approximations to bootstrap estimates. (English) Zbl 0548.62026

Beran, R.

Ann. Stat. 12, 101-118 (1984).

Let \(\hat T_ n=T(\hat F_ n)\), where \(\hat F_ n\) is the empirical df based on n i.i.d. observations \(X_ 1,...,X_ n\) with df F, and T is a locally quadratic functional defined on the set of all df’s on the real line whose support lies in a fixed compact interval. It is proved that the bootstrap estimators of the bias, variance and skewness of \(\hat T_ n\) are consistent; in addition these bootstrap estimators are shown to be asymptotically equivalent with their jackknife counterparts.
Also, the bootstrap distribution estimator of the df of \(n^{1/2}(\hat T_ n-T(F))\) is shown to be asymptotically equivalent to the one-term estimated Edgeworth expansion (i.e. with the coefficients appearing in the Edgeworth expansion estimated by jackknifing). Both distribution estimates also share the optimality property of being asymptotically minimax.

Reviewer: R.Helmers

MSC:

62G05	Nonparametric estimation
62E20	Asymptotic distribution theory in statistics
60F05	Central limit and other weak theorems

Keywords:

jackknife approximations; consistency; empirical distributions; locally quadratic functional; bootstrap estimators; bias; variance; skewness; Edgeworth expansion; asymptotically minimax