  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.
