UU-max-statistics

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 UU-statistics, defined as the average value of a real-valued kk-variate function hh calculated at all possible sets of kk points from a random sample. In the present paper, we investigate the corresponding extreme value analogue which we shall call UU-max-statistics. We are concerned with the behavior of the largest value of such a function hh instead of its average. Examples of UU-max-statistics are the diameter or the largest scalar product within a random sample. UU-max-statistics of higher degrees are given by triameters and other metric invariants.