Limit Theorems for U-Statistics

This chapter is also located at the interface between probability theory and statistics. It deals with U-statistics, which have numerous applications. In the one-sample case, a U-statistic averages a function of collections of k out of n random random vectors. These random vectors are independent, and they have the same distribution function F. A U-statistic is an unbiased estimator of a statistical functional \(\vartheta (F)\) . The first main result is a central limit theorem for so-called non-degenerate U-statistics, which is obtained using the Hájek projection. A mathematically deeper result is the limit distribution of U-statistics that have a degeneracy of first order. The proof uses a variant of the Hájek projection and the expansion theorem for linear self-adjoint compact operators from functional analysis. The chapter also contains a central limit theorem for two-sample U-statistics. Statistical applications include the Mann-Whitney U-test and the Cramér-von Mises goodness-of-fit test. The chapter concludes by highlighting the close relationship between U-statistics and the V-statistics, which are named after R. von Mises.