The Properties And Applications For The Estimator Of Quantiles

Quantile estimator is an important research topic in statistics, which was extensivelyapplied in many fields. Particularly, it was of great significance in the financial industryand medical industry. Quantile is an important risk measure in financial risk managementof financial, such as VaR and CVaR. As we know, VaR is a risk measure of value, usuallyit is quantile point of loss variable at given probability level, while CVaR is the mean ofthe losses that is greater than VaR. Therefore, VaR directly related with the quantileestimator, so the research of Quantile estimator is very significant.Scholars usually use the methods of parameter estimation and nonparametricestimation to research quantile estimators. However, nonparametric estimation dose notneed to assume the population distribution and build another model, so it is moreconvenient. Large numbers of scholars had done a lot of work on the nonparametricestimation of quantile estimator, such as KL quantile estimator, product quantile estimator,and the weighted order statistic quantile estimator. This article focused on the study of theproperties and applications of quantile estimator, the main research contents are asfollows:(1) The large deviation of the smoothed estimate of sample quantile. At first, thedefinition for kernel estimator of sample quantile, assumptions and several lemmas wereintroduced. Then the method of Gartner Ellistheorem was used to obtain thepointwise moderate deviations principle for theξ pn ξpbased on the identicallydistributed sample.(2) The moderate deviation of the smoothed estimate of sample quantile. At first, themoderate deviation ofξ pn ξpwas discussed. Then the special cases for the moderatedeviation of sample quantile was given, the cases proposed that the performance of kernelestimate properties were determined by the kernel function..(3)The application of quantile estimator. The questions about minimizing the totalweighted tardiness in single machine scheduling problem with release time and uncertainty processing time was reseached in order to introduce the applications of theestimator of quantile. The single machine scheduling problem was considered in this paper.By generating a set of scenarios for the processing time with some distribution;astochastic programming model minimizing the total weighted tardiness was formulated.The effectiveness of the proposed model has been verified by real instances.In this article;considered the pointwise large and moderate deviations principle fortheξ pn ξpwas obtained based on the identically distributed sample. Considered thesingle machine schedule-ng problem;a stochastic programming model was formulated topropose the effectiveness of the quantile estimator application model in control problem.This properties and applications lay the foundation for the researching properties andapplications of quantile estimator.