Archive
Special Issues 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
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
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
Reference

Andrew, E. B., et al. Conditional value-at-risk in portfolio optimization: Coherent but fragile. Operations Research Letters, 2011, 39: 163–171.

Artzner, P., Delbaen, F., Eber, J., Heath, D. Thinking coherently. Risk, 1997, 10: 68-71.

Artzner, P., Delbaen, F., Jean-Marceber. Coherent measues of risk. Mathematical Finance, 1999, 9 (3): 203–228.

Chen, H., et al. Conditional value-at-risk-based optimal spinning reserve for wind integrated power system. Int. Trans. Electr. Energ. Syst., 2016, 26: 1799–1809.

Escanciano, J. C., Mayoral, S. Semiparametric estimation of dynamic conditional expected shortfall models. Int. J. Monetary Economics and Finance, 2008, 1 (2): 106.

Föllmer, H., Schied, A. Stochastic Finance: An Introduction in Discrete Time. Walter De Gruyter Incorporated, 2004.

Gotoh, J., Takano, Y. Newsvendor solutions via conditional value-at-risk minization European. Journal of Operational Reserch, 2007, 170 (1): 80-96.

Legg, S. W., et al. Optimal gas detector placement under uncertainty considering Conditional Value-at-Risk. Journal of Loss Prevention in the Process Industries, 2013, 26: 410-417.

Lehmann, E. L. Some Concepts of Dependence. The Annals of Mathematical Statistics, 1966, 37 (5): 1137-1153.

Luo, Z., Yang, S. The Asymptotic properties of CVaR estimator under ρ mixing sequences. Acta Mathematica Sinica (Chinese Series), 2013, 56 (6): 851-870.

Luo, Z., Ou, S. The almost sure convergence rate of the estimator of optimized certainty equivalent risk measure under α-mixing sequences. Communications in Statistics-Theory and Methods, 2017, 46 (16): 8166–8177.

Mansini, R., Ogryczak, W., Speranza, M. G. Conditional value at risk and related linear programming models for portfolio optimization. Ann Oper Res, 2007, 152: 227–256.

Noyan, N., Rodulf, G. Optimization with Multivariate Conditional Value-at-Risk Constraints. operations research, 2013, 61 (4): 990–1013.

Pflug, G. Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk. Probabilistic Constrained Optimization, 2000: 272-277.

Rochafellar, R. T., Uryasev, S. Optimization of Conditional value-at-risk. The Journal of Risk, 2000, 3 (2): 21-24.

Rockafellar, R. T., Uryasev, S. Conditional value-at-risk for general loss distributions. Research report, 2002.

Roussas, G. G. Asymptotic normality of the kernel estimate of a probability density function under association. Statistics Probability Letters, 2000, 50: 1–12.

Scaillet, O. Nonparametric estimation and sensitivity ananlysis of expected shortfall. Mathematical Finance, 2004, 14 (1): 115–129.

Shapiro, A. Statistical inference of stochastic optimization problem. In: Uryasev. S. (ed), Probabilistic Constrained Optimization: Methodology and Applications, Kluwer Academic Publisher, Dordrecht, 2000: 282-304.

Su, C., Zhao, L. Moment inequalities and weak convergence for negatively associated sequences. Science in China (Series A), 1997, 40 (2): 172-182.

Su, C., Wang, Y. Strong Convergence for IDNA Sequences. Chiness Journal of Applied Probability and Statistics, 1998, 14 (2): 131-140.

Sun, H., et al. Optimal multivariate quota-share reinsurance: A nonparametric mean-CVaR framework. Insurance: Mathematics and Economics, 2017, 72: 197–214.

Trindade, A. A., et al. Financial prediction with constrained tail risk. Journal of Banking and Finance, 2007, 31 (11): 3524-3538.

Wang, C. S, Zhao, Z. Conditional Value-at-Risk: Semiparametric estimation and inference. Journal of Econometrics, 2016, 195: 86–103.

Xing, G, D. et al. Strong Consistency of Conditional Value-at-risk Estimate for ϕ-mixing Samples. Communications in Statistics-Theory and Methods, 2014, 43: 5105-5113.

Yang, S. Moment inequalities for partial sums of random variables. Science in China (Series A), 2001, 44: 1-6.

Yang, S. Uniformly asymptotic normality of the regression weighted estimator for negatively associated samples. Statistics-Probability letters, 2003: 101-110.

Zhou, X., et al. Moment consistency of estimators in partially linear models under NA samples. Metrika, 2010, 72: 415–432. 