Generalized two-sample U-statistics for clustered data

In this paper we investigate two-sample U-statistics in the case of clusters of repeated measurements observed on individuals from independent populations. The observations on the i-th individual in the first population are denoted by , 1 ≤ i ≤ m, and those on the k-th individual in the second population are denoted by , 1 ≤ k ≤ n. Given the kernel φ(x, y), we define the generalized two-sample U-statistic by We derive the asymptotic distribution of Um,n for large sample sizes. As an application we study the generalized Mann–Whitney–Wilcoxon rank sum test for clustered data.