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

TL;DR

This paper develops a new, general asymptotic theory for trimmed U-statistics using generalized L-statistics, relaxing continuity assumptions and providing improved conditions for asymptotic normality.

Contribution

It introduces a novel asymptotic result for generalized L-statistics that does not require distribution continuity at truncation points.

Findings

Provides a unified asymptotic distribution framework for trimmed U-statistics.

Relaxes the continuity requirement at truncation points.

Offers improved conditions for asymptotic normality.

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 $U$-statistic summands. This approach leads to improved conditions for the asymptotic normality of these trimmed $U$-statistics.