  Extreme U-statistics arise when the kernel of a U-statistic has a high degree
but depends only on its arguments through a small number of top order
statistics. As the kernel degree of the U-statistic grows to infinity with the
sample size, estimators built out of such statistics form an intermediate
family in between those constructed in the block maxima and
peaks-over-threshold frameworks in extreme value analysis. The asymptotic
normality of extreme U-statistics based on location-scale invariant kernels is
established. Although the asymptotic variance coincides with the one of the
H\'ajek projection, the proof goes beyond considering the first term in
Hoeffding's variance decomposition. We propose a kernel depending on the three
highest order statistics leading to a location-scale invariant estimator of the
extreme value index resembling the Pickands estimator. This extreme Pickands
U-estimator is asymptotically normal and its finite-sample performance is
competitive with that of the pseudo-maximum likelihood estimator.
