Volume 8, Issue 6, November 2019, Page: 253-260
Asymptotic Properties of Optimized Type CVaR Estimator for NA Random Variables
Shanchao Yang, School of Mathematics and Statistics, Guangxi Normal University, Guilin, China
Yuting Wang, School of Mathematics and Statistics, Guangxi Normal University, Guilin, China
Xin Yang, School of Mathematical Sciences, Guilin University of Aerospace Technology, Guilin, China
Xiutao Yang, Department of Basic Course Teaching, Haikou College of Economics, Haikou, China
Received: Oct. 8, 2019;       Accepted: Nov. 9, 2019;       Published: Nov. 25, 2019
DOI: 10.11648/j.ajtas.20190806.17      View  27      Downloads  18
Abstract
VaR and CVaR are important risk measures, which are widely used in finance, economy, insurance and other fields. However, VaR is not a coherent risk quantity, and it is not sufficient to measure tail risk. CVaR (also known as expected shortfall, ES) is a coherent risk measure, and it makes up for the defect that VaR is not enough to measure tail risk. Therefore, CVaR has been paid more and more attention in both application and theory fields. Rockafellar and Uryasev (2000) and Trindade et al (2007) proposed an optimized type CVaR estimator and studied some asymptotic properties of the estimator. Since then, some scholars have discussed the properties of the estimator in the cases of ρ-mixing, φ-mixing and α-mixing. In this paper, we shall study the asymptotic properties of the optimized type CVaR estimator in the case where the samples are NA random variables. The consistency and the asymptotic normality of the optimized type CVaR estimator and their corresponding convergence rates are obtained. The convergence rates of estimation are n-1/2 or near to n-1/2. These results also establish the asymptotic relations of the optimized type CVaR estimator and the common CVaR estimator. And their deviation converges almost surely to 0 at the rate of n-1/2.
Keywords
CVaR Estimator, Consistency, Asymptotic Normality, Convergence Rate, NA Sequence
To cite this article
Shanchao Yang, Yuting Wang, Xin Yang, Xiutao Yang, Asymptotic Properties of Optimized Type CVaR Estimator for NA Random Variables, American Journal of Theoretical and Applied Statistics. Vol. 8, No. 6, 2019, pp. 253-260. doi: 10.11648/j.ajtas.20190806.17
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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