Cram\'{e}r type large deviations for trimmed L-statistics

TL;DR

This paper introduces a new method to analyze the large deviation probabilities of trimmed L-statistics by approximating them with non-trimmed L-statistics based on Winsorized variables, under mild conditions.

Contribution

It presents a novel approach to study large deviations of trimmed L-statistics using approximation by Winsorized non-trimmed L-statistics, improving on previous restrictive conditions.

Findings

Established Cramér type large deviation results for trimmed L-statistics.

Provided a more natural and less restrictive set of conditions for analysis.

Extended the understanding of large deviations in robust statistical measures.

Abstract

In this paper, we propose a new approach to the investigation of asymptotic properties of trimmed $L$-statistics and we apply it to the Cram\'{e}r type large deviation problem. Our results can be compared with ones in Callaert et al.(1982) -- the first and, as far as we know, the single article, where some results on probabilities of large deviations for the trimmed $L$-statistics were obtained, but under some strict and unnatural conditions. Our approach is to approximate the trimmed $L$-statistic by a non-trimmed $L$-statistic (with smooth weight function) based on Winsorized random variables. Using this method, we establish the Cram\'{e}r type large deviation results for the trimmed $L$-statistics under quite mild and natural conditions.