Bootstrap Approximation to Distributions of Finite Population U-statistics

We show that the without replacement bootstrap of Booth, Butler and Hall (J. Am. Stat. Assoc. 89, 1282–1289, 1994) provides second order correct approximation to the distribution function of a Studentized U-statistic based on simple random sample drawn without replacement. In order to achieve similar approximation accuracy for the bootstrap procedure due to Bickel and Freedman (Ann. Stat. 12, 470–482, 1984) and Chao and Lo (Sankhya Ser. A 47, 399–405, 1985) we introduce randomized adjustments to the resampling fraction.