One-term Edgeworth expansion for finite population U-statistics of degree two

By means of Hoeffding's decomposition, we represent a finite population U-statistic of degree two by the sum of a linear and a quadratic part. Assuming that the linear part is nondegenerate, we prove the validity of one-term Edgeworth expansion for the distribution function of the statistic under the optimal (minimal) conditions on the linear part and 2 + delta moment condition on the quadratic part. No condition is imposed on the ratio N / n, where N, respectively n, denotes the sample size respectively the population size.