Title: Cramer type large and moderate deviations for trimmed $L$-statistics
Authors: Nadezhda Gribkova - Saint-Petersburg State University (Russia) [presenting]
Abstract: The class of $L$-statistics is one of the most commonly used classes in statistical inferences; the famous Gini index also belongs to this class. There is an extensive literature on asymptotic properties of $L$-statistics, but its part related to large deviations is not so vast. Only a few highly sharp results on large deviations for non-trimmed $L$-statistics with coefficients generated by a smooth on $(0,1)$ weight function are mentioned. These results, however, do not cover the case of trimmed $L$-statistics, i.e., the case when the weight function is zero outside of some interval $[\alpha,1-\beta]\subset (0,1)$. Our recent results on Cramer type large and moderate deviations for trimmed $L$-statistics will be presented, and our approach for solving this problem will be discussed. This approach is to approximate the trimmed $L$-statistic by a non-trimmed $L$-statistic with coefficients generated by a smooth on $(0,1)$ weight function, where the approximating $L$-statistic is based on order statistics corresponding to a sample of i.i.d. Winsorized random variables.