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.Comment: 17 page

Second Order Approximations for Slightly Trimmed Sums

We investigate the second order asymptotic behavior of trimmed sums T_n=\frac 1n \sum_{i=\kn+1}^{n-\mn}\xin, where \kn, \mn are sequences of integers, 0\le \kn < n-\mn \le n, such that \min(\kn, \mn) \to \infty, as \nty, the \xin's denote the order statistics corresponding to a sample $X_1,...,X_n$ of $n$ i.i.d. random variables. In particular, we focus on the case of slightly trimmed sums with vanishing trimming percentages, i.e. we assume that \max(\kn,\mn)/n\to 0, as \nty, and heavy tailed distribution $F$, i.e. the common distribution of the observations $F$ is supposed to have an infinite variance. We derive optimal bounds of Berry -- Esseen type of the order $O\bigl(r_n^{-1/2}\bigr)$, r_n=\min(\kn,\mn), for the normal approximation to $T_n$ and, in addition, establish one-term expansions of the Edgeworth type for slightly trimmed sums and their studentized versions. Our results supplement previous work on first order approximations for slightly trimmed sums by Csorgo, Haeusler and Mason (1988) and on second order approximations for (Studentized) trimmed sums with fixed trimming percentages by Gribkova and Helmers (2006, 2007).Comment: 37 pages, to appear in Theory Probab. App

Vector quantization and clustering in presence of censoring

We consider the problem of optimal vector quantization for random vectors with one censored component and applications to clustering of censored observations. We introduce the definitions of the empirical distortion and of the empirically optimal quantizer in presence of censoring and we establish the almost sure consistency of empirical design. Moreover, we provide a non asymptotic exponential bound for the difference between the performance of the empirically optimal k-quantizer and the optimal performance over the class of all k-quantizers. As a natural application of the new quantization criterion, we propose an iterative two-step algorithm allowing for clustering of multivariate observations with one censored component. This method is investigated numerically through applications to real and simulated data

On the M fewer than N bootstrap approximation to the trimmed mean

We show that the M fewer than N (N is the real data sample size, M denotes the size of the bootstrap resample; M=N ! 0, as M ! 1) bootstrap approximation to the distribution of the trimmed mean is consistent without any conditions on the population distribution F, whereas Efron's naive (i.e. M = N) bootstrap as well as the normal approximation fails to be consistent if the population distribution F has gaps at the two quantiles where the trimming occurs

Educational content of network is basis of interaction in system of additional vocational training

В статье обозначены проблемы эффективного использования сетевых электронных ресурсов в системе дополнительного профессионального образования, представлен краткий обзор полнотекстовых электронных ресурсов Института развития образования Сахалинской области.In article is presented short review of electronic resources of Institute of a development of education of the Sakhalin region and the problems, of effective utilisation of network in system of additional vocational trainin