An extreme U-statistic occurs when the kernel of the U-statistic has a high order, but depends only on arguments with a few top-level statistics. As the kernel order of the U statistic goes to infinity with sample size, the estimator constructed from such a statistic will be similar to that constructed in the block maxima and peaks above threshold frameworks of extreme value analysis. form an intermediate family. Asymptotic normality of the extreme U statistic based on the location scale invariant kernel is established. The asymptotic variance agrees with the variance of the H\’ajek projection, but the proof does not only consider the first term of Hoeffding’s variance decomposition. We propose a kernel that depends on the three highest-order statistics leading to a position-scale-invariant estimator of extreme index similar to the Pickands estimator. This extreme Pickands U estimator is asymptotically normal, and its finite-sample performance competes with the quasi-maximum-likelihood estimator.
