Joint inference on extreme expectiles for multivariate heavy-tailed distributions

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Expectiles induce a law-invariant, coherent and elicitable risk measure that has received substantial attention in actuarial and financial risk management contexts. A number of recent papers have focused on the behaviour and estimation of extreme expectile-based risk measures and their potential for risk assessment was highlighted in financial and actuarial real data applications. Joint inference of several extreme expectiles has however been left untouched; in fact, even the inference about a marginal extreme expectile turns out to be a difficult problem in finite samples, even though an accurate idea of estimation uncertainty is crucial for the construction of confidence intervals in applications to risk management. We investigate the joint estimation of extreme marginal expectiles of a random vector with heavy-tailed marginal distributions, in a general extremal dependence model. We use these results to derive corrected confidence regions for extreme expectiles, as well as a test for the equality of tail expectiles. The methods are showcased in a finite-sample simulation study and on real financial data.