Weighted bootstrap for two-sample U-statistics

In this paper, we introduce weighted bootstrap algorithms for both non-degenerate and degenerate two-sample $U$-statistics with arbitrary degrees. For the non-degenerate case, weighted bootstrap with dependent weights is introduced as a generalization of Efron’s conventional bootstrap. In addition, two weighted bootstrap procedures with independent productive weights and independent additive weights are proposed under non-degeneracy. More importantly, we extend the weighted bootstrap method to two-sample $U$-statistics under the degeneracy of order 2 with a novel construction of random weights. Theoretical supports of the proposed weighted bootstrap procedures under non-degeneracy and degeneracy of order 2 are established. Numerical studies illustrate that the proposed approaches are feasible and effective for statistical inferences.