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

TL;DR

This paper studies the asymptotic behavior of multivariate U-statistics with estimated nuisance parameters, establishing Gaussian limits under null and local alternatives, extending previous work to multivariate data and local alternatives.

Contribution

It extends Randles' work by analyzing multivariate U-statistics with estimated nuisance parameters under local alternatives and quantifies the impact of knowing some nuisance parameters.

Findings

Asymptotic Gaussian distribution under null hypothesis

Extension to multivariate data and kernels

Behavior under local alternatives analyzed for the first time

Abstract

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 provided 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 two specific examples.