Jackknife Variance Estimation for Hájek-Dominated Generalized U-Statistics

Abstract: We prove ratio-consistency of the jackknife variance estimator, and certain variants, for a broad class of generalized U-statistics whose variance is asymptotically dominated by their H\'ajek projection, with the classical fixed-order case recovered as a special instance. This H\'ajek projection dominance condition unifies and generalizes several criteria in the existing literature, placing the simple nonparametric jackknife on the same footing as the infinitesimal jackknife in the generalized setting. As an illustration, we apply our result to the two-scale distributional nearest-neighbor regression estimator, obtaining consistent variance estimates under substantially weaker conditions than previously required.