Exponential finite sample bounds for incomplete U-statistics

Incomplete U-statistics have been proposed to accelerate computation. They use only a subset of the subsamples required for kernel evaluations by complete U-statistics. This paper gives a finite sample bound in the style of Bernstein's inequality. Applied to complete U-statistics the resulting inequality improves over the bounds of both Hoeffding and Arcones. For randomly determined subsamples it is shown, that, as soon as the their number reaches the square of the sample-size, the same order bound is obtained as for the complete statistic.