### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 198-205 |

Number of pages | 8 |

Journal | Insurance: Mathematics and Economics |

Volume | 47 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2010 Oct 1 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Economics and Econometrics
- Statistics, Probability and Uncertainty

### Cite this

}

*Insurance: Mathematics and Economics*, vol. 47, no. 2, pp. 198-205. https://doi.org/10.1016/j.insmatheco.2010.05.001

**Bias correction for estimated distortion risk measure using the bootstrap.** / Kim, Joseph H.T.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Bias correction for estimated distortion risk measure using the bootstrap

AU - Kim, Joseph H.T.

PY - 2010/10/1

Y1 - 2010/10/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77955664132&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955664132&partnerID=8YFLogxK

U2 - 10.1016/j.insmatheco.2010.05.001

DO - 10.1016/j.insmatheco.2010.05.001

M3 - Article

VL - 47

SP - 198

EP - 205

JO - Insurance: Mathematics and Economics

JF - Insurance: Mathematics and Economics

SN - 0167-6687

IS - 2

ER -