The large-sample behavior of non-degenerate multivariate $U$-statistics of arbitrary degree is investigated under the assumption that their kernel depends on parameters that can be estimated consistently. Mild regularity conditions are given which guarantee that once properly normalized, such statistics are asymptotically multivariate Gaussian both under the null hypothesis and sequences of local alternatives. The work of Randles (1982, Ann. Statist.) is extended in three ways: the data and the kernel values can be multivariate rather than univariate, the limiting behavior under local alternatives is studied for the first time, and the effect of knowing some of the nuisance parameters is quantified. These results can be applied to a broad range of goodness-of-fit testing contexts, as shown in one specific example.

翻译：研究了在核函数依赖于可一致估计参数的假设下，任意阶非退化多元 $U$-统计量的大样本行为。给出了温和的正则条件，确保这些统计量在适当标准化后，在零假设和局部备择序列下均渐近服从多元高斯分布。Randles（1982, Ann. Statist.）的工作在三个方面得到扩展：数据和核值可为多元而非一元；首次研究了局部备择假设下的极限行为；量化了已知部分 nuisance 参数的影响。这些结果可广泛应用于拟合优度检验场景，具体示例中予以展示。