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 P₀ be a fixed probability measure on (X, B). We consider limiting distribution theory for U-statistics Tₙ = n⁻¹ ∑i ≠ j Q(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 (∫ 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.