Reduced $U$-Statistics and the Hodges-Lehmann Estimator

Abstract

A reduced $U$-statistic (of order 2) is defined as the sum of terms $f(X_i, X_j),$ where $f$ is a symmetric function, $(X_1, \cdots, X_N)$ are independent and identically distributed (i.i.d.) random variables (rv's), and $(i,j)$ are drawn from a restricted, though balanced, set of pairs. (A $U$-statistic corresponds to summation over all $(i, j)$ pairs.) A limit normal distribution is found for the reduced $U$-statistic, and it follows that estimates based on reduced $U$-statistics can have asymptotic efficiencies comparable with those based on $U$-statistics, even though the number of steps in computing a reduced $U$-statistic becomes asymptotically negligible in comparison with the number required for the corresponding $U$-statistic. As an illustration, a short-cut version of the Hodges-Lehmann estimator is defined, and its asymptotic properties derived, from a corresponding reduced $U$-statistic. A multivariate limit theorem is proved for a vector of reduced $U$-statistics, plus another result obtaining asymptotic normality even when $(i, j)$ are drawn from an unbalanced set of pairs. The present results are related to those of Blom.