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

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.