In 1948, W. Hoeffding [W. Hoeffding, A class of statistics with asymptotically normal distribution, Ann. Math. Statist. 19 (1948) 293-325] introduced a large class of unbiased estimators called U-statistics, defined as the average value of a real-valued k-variate function h calculated at all possible sets of k points from a random sample. In the present paper, we investigate the corresponding extreme value analogue which we shall call U-max-statistics. We are concerned with the behavior of the largest value of such a function h instead of its average. Examples of U-max-statistics are the diameter or the largest scalar product within a random sample. U-max-statistics of higher degrees are given by triameters and other metric invariants. © 2008 Elsevier Inc. All rights reserved.
Lao, W., & Mayer, M. (2008). U-max-statistics. Journal of Multivariate Analysis, 99(9), 2039–2052. https://doi.org/10.1016/j.jmva.2008.02.001