Asymptotic Properties of Optimized Type CVaR Estimator for NA Random Variables

Shanchao Yang; Yuting Wang; Xin Yang; Xiutao Yang

doi:doi:10.11648/j.ajtas.20190806.17

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Asymptotic Properties of Optimized Type CVaR Estimator for NA Random Variables

Shanchao Yang, Yuting Wang, Xin Yang, Xiutao Yang

Published in American Journal of Theoretical and Applied Statistics (Volume 8, Issue 6)

Received: 8 October 2019 Accepted: 9 November 2019 Published: 25 November 2019

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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.

Published in	American Journal of Theoretical and Applied Statistics (Volume 8, Issue 6)
DOI	10.11648/j.ajtas.20190806.17
Page(s)	253-260
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

CVaR Estimator, Consistency, Asymptotic Normality, Convergence Rate, NA Sequence

References

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APA Style

Shanchao Yang, Yuting Wang, Xin Yang, Xiutao Yang. (2019). Asymptotic Properties of Optimized Type CVaR Estimator for NA Random Variables. American Journal of Theoretical and Applied Statistics, 8(6), 253-260. https://doi.org/10.11648/j.ajtas.20190806.17

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ACS Style

Shanchao Yang; Yuting Wang; Xin Yang; Xiutao Yang. Asymptotic Properties of Optimized Type CVaR Estimator for NA Random Variables. Am. J. Theor. Appl. Stat. 2019, 8(6), 253-260. doi: 10.11648/j.ajtas.20190806.17

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AMA Style

Shanchao Yang, Yuting Wang, Xin Yang, Xiutao Yang. Asymptotic Properties of Optimized Type CVaR Estimator for NA Random Variables. Am J Theor Appl Stat. 2019;8(6):253-260. doi: 10.11648/j.ajtas.20190806.17

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@article{10.11648/j.ajtas.20190806.17,
  author = {Shanchao Yang and Yuting Wang and Xin Yang and Xiutao Yang},
  title = {Asymptotic Properties of Optimized Type CVaR Estimator for NA Random Variables},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {8},
  number = {6},
  pages = {253-260},
  doi = {10.11648/j.ajtas.20190806.17},
  url = {https://doi.org/10.11648/j.ajtas.20190806.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20190806.17},
  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.},
 year = {2019}
}

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TY - JOUR
T1 - Asymptotic Properties of Optimized Type CVaR Estimator for NA Random Variables
AU - Shanchao Yang
AU - Yuting Wang
AU - Xin Yang
AU - Xiutao Yang
Y1 - 2019/11/25
PY - 2019
N1 - https://doi.org/10.11648/j.ajtas.20190806.17
DO - 10.11648/j.ajtas.20190806.17
T2 - American Journal of Theoretical and Applied Statistics
JF - American Journal of Theoretical and Applied Statistics
JO - American Journal of Theoretical and Applied Statistics
SP - 253
EP - 260
PB - Science Publishing Group
SN - 2326-9006
UR - https://doi.org/10.11648/j.ajtas.20190806.17
AB - 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.
VL - 8
IS - 6
ER -

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Author Information

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

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APA Style

Shanchao Yang, Yuting Wang, Xin Yang, Xiutao Yang. (2019). Asymptotic Properties of Optimized Type CVaR Estimator for NA Random Variables. American Journal of Theoretical and Applied Statistics, 8(6), 253-260. https://doi.org/10.11648/j.ajtas.20190806.17

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ACS Style

Shanchao Yang; Yuting Wang; Xin Yang; Xiutao Yang. Asymptotic Properties of Optimized Type CVaR Estimator for NA Random Variables. Am. J. Theor. Appl. Stat. 2019, 8(6), 253-260. doi: 10.11648/j.ajtas.20190806.17

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AMA Style

Shanchao Yang, Yuting Wang, Xin Yang, Xiutao Yang. Asymptotic Properties of Optimized Type CVaR Estimator for NA Random Variables. Am J Theor Appl Stat. 2019;8(6):253-260. doi: 10.11648/j.ajtas.20190806.17

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@article{10.11648/j.ajtas.20190806.17,
  author = {Shanchao Yang and Yuting Wang and Xin Yang and Xiutao Yang},
  title = {Asymptotic Properties of Optimized Type CVaR Estimator for NA Random Variables},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {8},
  number = {6},
  pages = {253-260},
  doi = {10.11648/j.ajtas.20190806.17},
  url = {https://doi.org/10.11648/j.ajtas.20190806.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20190806.17},
  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.},
 year = {2019}
}

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