An Edgeworth Expansion for $U$-Statistics Based on Samples from Finite Populations

Abstract

Suppose that $U$ is a $U$-statistic of degree 2 based on a simple random sample of size $n$ selected without replacement from a finite population of $N$ elements. A bound for the difference between the distribution function of a standardized version of $U$ and its single-term Edgeworth expansion is given. We apply these results to obtain an Edgeworth expansion for the variance estimator in a finite population. Some simulation results are reported in this case.