Jackknife Variance Estimation for Hájek-Dominated Generalized U-Statistics

Abstract

Valid uncertainty quantification for subsampling-based and randomized estimators often depends on variance estimators whose behavior is much less understood than that of the underlying point estimator. We prove ratio-consistency of the jackknife variance estimator, and certain delete-$d$ variants, for a broad class of generalized U-statistics whose variance is asymptotically dominated by their Hajek projection and whose normalized first-projection squares satisfy a row-wise $L^r$ weak law, with the classical fixed-order case recovered as a special instance. This projection-dominance plus square-LLN structure unifies and generalizes several criteria from the existing literature, clarifies when the simple nonparametric jackknife is theoretically justified in the generalized setting, and yields consistent variance estimation for the two-scale distributional nearest-neighbor regression estimator under substantially weaker conditions than previously required.