U-Quantile-Statistics

In 1948, W. Hoeffding introduced a large class of unbiased estimators called U-statistics, defined as the average value of a real-valued m-variate function h calculated at all possible sets of m points from a random sample. In the present paper, we investigate the corresponding robust analogue which we call U-quantile-statistics. We are concerned with the asymptotic behavior of the sample p-quantile of such function h instead of its average. Alternatively, U-quantile-statistics can be viewed as quantile estimators for a certain class of dependent random variables. Examples are given by a slightly modified Hodges-Lehmann estimator of location and the median interpoint distance among random points in space.

💡 Research Summary

The paper introduces “U‑quantile‑statistics,” a robust counterpart to the classical U‑statistics originally formulated by Hoeffding in 1948. While a U‑statistic is defined as the average of a symmetric m‑variate kernel h evaluated over all possible m‑tuples drawn from an i.i.d. sample, the authors replace the averaging operation with the sample p‑quantile of the collection of h‑values. This simple modification yields estimators that retain the desirable asymptotic properties of U‑statistics—such as unbiasedness under symmetry and a well‑understood Hoeffding decomposition—while dramatically improving resistance to outliers.

The authors first formalize the object of interest: let F_n be the empirical distribution function of the N = C(n,m) kernel values h(X_{i1},…,X_{im}). The U‑quantile statistic Q_n(p) is defined as the smallest t for which F_n(t) ≥ p, i.e., the sample p‑quantile of the kernel values. Under mild regularity conditions (continuity of the underlying distribution at the true quantile θ_p, existence of a positive density f(θ_p), and finite 2k‑th moments of the kernel), they derive a Bahadur‑Kiefer type representation: \