ASYMPTOTIC MEAN SQUARE ERRORS OF VARIANCE ESTIMATORS FOR U-STATISTICS AND THEIR EDGEWORTH EXPANSIONS

Abstract

This paper studies the variance estimators for a class of U-statistics. We obtain asymptotic representations of jackknife, Hinkley's (1978) corrected jackknife, unbiased, Sen's (1960), and new variance estimators. The asymptotic mean square errors of these estimators are theoretically investigated. The Edgeworth expansions of the estimators with remainder term o(n^-1) are also established. We show that the normalized Hinkley's corrected estimator coincides with the normalized unbiased estimator up until the order n^-1/2o_p(n^-1).