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The large-sample behavior of non-degenerate multivariate $U$-statistics ofarbitrary degree is investigated under the assumption that their kernel dependson parameters that can be estimated consistently. Mild regularity conditionsare given which guarantee that once properly normalized, such statistics areasymptotically multivariate Gaussian both under the null hypothesis andsequences of local alternatives. The work of Randles (1982, Ann. Statist.) isextended in three ways: the data and the kernel values can be multivariaterather than univariate, the limiting behavior under local alternatives isstudied for the first time, and the effect of knowing some of the nuisanceparameters is quantified. These results can be applied to a broad range ofgoodness-of-fit testing contexts, as shown in one specific example.

Asymptotics for non-degenerate multivariate $U$-statistics with estimated nuisance parameters under the null and local alternative hypotheses