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.

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关于 U 统计量方差估计量的渐近行为

摘要 U 统计量具有良好的性质，例如渐近正态性、无偏性和无偏估计量之间的最小方差。它们的方差的估计通常很有趣，例如导出渐近检验。众所周知，U 统计量方差的无偏估计量可以明确地表述为 U 统计量本身，但对样本大小的特定依赖性使渐近陈述变得困难。在这里，我们通过将方差估计量分解为具有固定核大小的 U 统计量的线性组合来解决该问题，从而获得关于渐近分布的直接陈述。我们随后证明了学生化估计量的中心极限定理。我们表明，它导致了一个假设检验，该检验比较了两种预测算法的误差估计，并允许为误差的真实差异构建渐近精确的置信区间。该测试通过实际数据应用和模拟研究来说明。