Large Sample Theory for U-Statistics and Tests of Fit

Let X₍₈, i = 1, , n be i. i. d. random variables on an arbitrary measurable space (X, B). Suppose L (X₍₈) = Q₍₁, i = 1, , n and let P₀ be a fixed probability measure on (X, B). We consider limiting distribution theory for U-statistics Tₙ = n^-1 ₈ ₉ Q (X₍₈, X₍₉) (1) under conditions which imply the product measures Qₙ = Q₍₁ Q₍₁, n times, are contiguous to the product measures Pₙ = P₀ P₀, n times, and (2) for kernels Q which are symmetric, square-integrable (Q² (, ) dP₀ P₀ <) and degenerate in a certain sense (Q (, t) P₀ (dt) = 0 a. e. (P₀) ). 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.