Stute [Ann. Probab. 19 (1991) 812–825] introduced a class of estimators called conditional U-statistics of $\mathbb{E}(\phi (Y_1,\dots ,Y_m)\mid (X_1,\dots ,X_m)=\mathbf{t}),\text{for }\mathbf{t}\in {\mathbb{R}}^m.$ 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.

斯图特 [安. 可能。19 (1991) 812-825] 引入了一类称为条件U统计量的估计量$\mathbb{乙}(\phi ({是}_1,\dots ,{是}_{米})\mid (X_1,\dots ,X_{米})=\mathbf{吨}),\text{为了 }\mathbf{吨}\in {\mathbb{电阻}}^{米}.$在目前的工作中，我们提供了一类新的条件U统计估计量。更准确地说，我们研究了基于 copula 表示的条件U统计量。我们为建议的估计器建立了均匀带宽一致性。我们的定理允许数据驱动的本地带宽用于这些统计数据。本文中建立的理论一致一致性结果是（或将是）copula 回归分析中许多进一步发展的关键工具。这些程序的性能是通过在条件 Kendall's tau 的上下文中进行模拟来评估的。