Asymptotic distributions for a class of generalized $L$-statistics

Abstract

We adapt the techniques in Stigler [Ann. Statist. 1 (1973) 472-477] to obtain a new, general asymptotic result for trimmed U -statistics via the generalized L-statistic representation introduced by Serfling [Ann. Statist. 12 (1984) 76-86]. Unlike existing results, we do not require continuity of an associated distribution at the truncation points. Our results are quite general and are expressed in terms of the quantile function associated with the distribution of the Ustatistic summands. This approach leads to improved conditions for the asymptotic normality of these trimmed U -statistics.

Non-uniform Berry–Esseen Bounds for Weighted U-Statistics and Generalized L-Statistics

“…The asymptotic properties of the generalized L-statistics have been extensively studied. We refer to Serfling [12,13] Borovshikh and Weber [1] for the asymptotic normality and for the general theory and applications, Cai [2] for the moderate and large deviations, and Helmers and Ruymgaart [6], Helmers, Jansseen and Serfling [5] for the uniform Berry-Esseen bounds. Let H (y) be the distribution function of the random variable function h(X 1 , .…”