Asymptotic normality for U-statistics based on trimmed samples

Abstract

Let Xn1 ≤ cdots, three dots, centered ≤ Xnn be an ordered sample of size n. We establish asymptotic normality of U-statistics based on the trimmed sample Xn,[αn]+1≤ cdots, three dots, centered ≤ Xn,n − [βn] where 0<α, β<1/2. This theorem and its multi-sample generalization are illustrated by various statistics of importance for robust estimation of location, dispersion, etc. This unifies the flexibility of the class of U-statistics and the classical principle of rejection of outliners.