Bias correction for estimated distortion risk measure using the bootstrap
The bias of the empirical estimate of a given risk measure has recently been of interest in the risk management literature. In particular, Kim and Hardy (2007) showed that the bias can be corrected for the Conditional Tail Expectation (CTE, a.k.a. Tail-VaR or Expected Shortfall) using the bootstrap. This article extends their result to the distortion risk measure (DRM) class where the CTE is a special case. In particular, through the exact bootstrap, it is analytically proved that the bias of the empirical estimate of DRM with concave distortion function is negative and can be corrected on the bootstrap, using the fact that the bootstrapped loss is majorized by the original loss vector. Since the class of DRM is a subset of the L-estimator class, the result provides a sufficient condition for the bootstrap bias correction for L-estimators. Numerical examples are presented to show the effectiveness of the bootstrap bias correction. Later a practical guideline to choose the estimate with a lower mean squared error is also proposed based on the analytic form of the double bootstrapped estimate, which can be useful in estimating risk measures where the bias is non-cumulative across loss portfolio.
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- Jones, Bruce L. & Zitikis, Ricardas, 2007. "Risk measures, distortion parameters, and their empirical estimation," Insurance: Mathematics and Economics, Elsevier, vol. 41(2), pages 279-297, September.
- Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
- Inui, Koji & Kijima, Masaaki & Kitano, Atsushi, 2005. "VaR is subject to a significant positive bias," Statistics & Probability Letters, Elsevier, vol. 72(4), pages 299-311, May.
- Kim, Joseph H.T. & Hardy, Mary R., 2009. "Estimating the Variance of Bootstrapped Risk Measures," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 39(01), pages 199-223, May.
- Kim, Joseph Hyun Tae & Hardy, Mary R., 2007. "Quantifying and Correcting the Bias in Estimated Risk Measures," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 37(02), pages 365-386, November.
- Jones, Bruce L. & Puri, Madan L. & Zitikis, Ricardas, 2006. "Testing hypotheses about the equality of several risk measure values with applications in insurance," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 253-270, April.
- A. D. Hutson & M. D. Ernst, 2000. "The exact bootstrap mean and variance of an "L"-estimator," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(1), pages 89-94.
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