Large Sample Theory for U-Statistics and Tests of Fit

Let Xni, i = 1, ⋯, n be i.i.d. random variables on an arbitrary measurable space (X, B). Suppose L(Xni) = Qn1, i = 1, ⋯, n and let P0be a fixed probability measure on (X, B). We consider limiting distribution theory for U-statistics Tn= n-1∑i ≠ jQ(Xni, Xnj) (1) under conditions which imply the product measures Qn= Qn1× ⋯ × Qn1, n times, are contiguous to the product measures Pn= P0× ⋯ × P0, 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.