Robust estimation of U-statistics

An important part of the legacy of Evarist Gin\'e is his fundamental contributions to our understanding of $U$-statistics and $U$-processes. In this paper we discuss the estimation of the mean of multivariate functions in case of possibly heavy-tailed distributions. In such situations, reliable estimates of the mean cannot be obtained by usual $U$-statistics. We introduce a new estimator, based on the so-called median-of-means technique. We develop performance bounds for this new estimator that generalizes an estimate of Arcones and Gin\'e (1993), showing that the new estimator performs, under minimal moment conditions, as well as classical $U$-statistics for bounded random variables. We discuss an application of this estimator to clustering.