On the uniform-in-bandwidth consistency of the general conditionalU-statistics based on the copula representation

Stute [Ann. Probab. 19 (1991) 812–825] introduced a class of estimators called conditional U-statistics of E(φ(Y1,…,Ym)∣(X1,…,Xm)=t),for t∈Rm. In the present work, we provide a new class of estimators of conditional U-statistics. More precisely, we investigate the conditional U-statistics based on copula representation. We establish the uniform-in-bandwidth consistency for the proposed estimator. Our theorems allow data-driven local bandwidths for these statistics. The theoretical uniform consistency results, established in this paper, are (or will be) key tools for many further developments in copula regression analysis. The performance of these procedures is evaluated through simulation in the context of the conditional Kendall's tau.