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Journal Articles Journal of Statistical Planning and Inference Year : 2024

Reduced-bias estimation of the extreme conditional tail expectation for Box-Cox transforms of heavy-tailed distributions

Abstract

Conditional tail expectation (CTE) is a coherent risk measure defined as the mean of the loss distribution above a high quantile. The existence of the CTE as well as the asymptotic properties of associated estimators however require integrability conditions that may be violated when dealing with heavy-tailed distributions. We introduce Box-Cox transforms of the CTE that have two benefits. First, they alleviate these theoretical issues. Second, they enable to recover a number of risk measures such as conditional tail expectation, expected shortfall, conditional value-at-risk or conditional tail variance. The construction of dedicated estimators is based on the investigation of the asymptotic relationship between Box-Cox transforms of the CTE and quantiles at extreme probability levels, as well as on an extrapolation formula established in the heavy-tailed context. We quantify and estimate the bias induced by the use of these approximations and then introduce reduced-bias estimators whose asymptotic properties are rigorously shown. Their finite-sample properties are assessed on a simulation study and illustrated on real data, highlighting the practical interest of both the bias reduction and the Box-Cox transform.
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Dates and versions

hal-04243953 , version 1 (16-10-2023)
hal-04243953 , version 2 (11-04-2024)

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  • HAL Id : hal-04243953 , version 2

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Michaël Allouche, Jonathan El Methni, Stéphane Girard. Reduced-bias estimation of the extreme conditional tail expectation for Box-Cox transforms of heavy-tailed distributions. Journal of Statistical Planning and Inference, inPress, pp.1-42. ⟨hal-04243953v2⟩
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