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Estimating the Variance of Bootstrapped Risk Measures

Published online by Cambridge University Press:  09 August 2013

Joseph H.T. Kim  and
Mary R. Hardy
Mary R. Hardy
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, CanadaN2L 3G1, E-Mail: mrhardy@math.uwaterloo.ca
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Abstract

In Kim and Hardy (2007) the exact bootstrap was used to estimate certain risk measures including Value at Risk and the Conditional Tail Expectation. In this paper we continue this work by deriving the influence function of the exact-bootstrapped quantile risk measure. We can use the influence function to estimate the variance of the exact-bootstrap risk measure. We then extend the result to the L-estimator class, which includes the conditional tail expectation risk measure. The resulting formula provides an alternative way to estimate the variance of the bootstrapped risk measures, or the whole L-estimator class in an analytic form. A simulation study shows that this new method is comparable to the ordinary resampling-based bootstrap method, with the advantages of an analytic approach.

Keywords

Exact bootstrapL-estimatorinfluence functionnonparametric delta methodvariance estimationdistortion risk measure
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 39 , Issue 1 , May 2009 , pp. 199 - 223
Copyright
Copyright © International Actuarial Association 2009

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