Large Sample Theory for $U$-Statistics and Tests of Fit

Abstract

Let $X_{ni}, i = 1, \cdots, n$ be i.i.d. random variables on an arbitrary measurable space $(\mathscr{X}, B)$. Suppose $\mathscr{L}(X_{ni}) = Q_{n1}, i = 1, \cdots, n$ and let $P_0$ be a fixed probability measure on $(\mathscr{X}, B)$. We consider limiting distribution theory for $U$-statistics $T_n = n^{-1} \sum_{i \neq j} Q(X_{ni}, X_{nj})$ (1) under conditions which imply the product measures $Q_n = Q_{n1} \times \cdots \times Q_{n1}, n$ times, are contiguous to the product measures $P_n = P_0 \times \cdots \times P_0, n$ times, and (2) for kernels $Q$ which are symmetric, square-integrable $(\int Q^2(\bullet, \bullet) dP_0 \times P_0 < \infty)$ and degenerate in a certain sense $(\int Q(\bullet, t)P_0(dt) = 0 \mathrm{a.e.} (P_0))$. Applications to chi-square and Cramer-von Mises tests for a simple hypothesis and Cramer-von Mises tests for the case when parameters have to be estimated, are given. A tail sensitive test for normality is introduced.