On the asymptotic behaviour of the variance estimator of a [formula omitted]-statistic

Abstract

U-statistics enjoy good properties such as asymptotic normality, unbiasedness and minimal variance among unbiased estimators. The estimation of their variance is often of interest, for instance to derive asymptotic tests. It is well-known that an unbiased estimator of the variance of a U-statistic can be formulated explicitly as a U-statistic itself, but specific dependencies on the sample size make asymptotic statements difficult. Here, we solve the issue by decomposing the variance estimator into a linear combination of U-statistics with fixed kernel size, consequently obtaining a straightforward statement on the asymptotic distribution. We subsequently demonstrate a central limit theorem for the studentized estimator. We show that it leads to a hypothesis test which compares the error estimates of two prediction algorithms and permits construction of an asymptotically exact confidence interval for the true difference of errors. The test is illustrated by a real data application and a simulation study.