Extreme expectile estimation for heavy-tailed time series
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Expectiles define a least squares analogue of quantiles. They have lately received substantial attention in actuarial and financial risk management contexts. Unlike quantiles, expectiles define coherent risk measures and are determined by tail expectations rather than tail probabilities; unlike the popular Expected Shortfall, they define elicitable risk measures. This has motivated the study of the behaviour and estimation of extreme expectiles in some of the recent statistical literature. The case of stationary but weakly dependent observations has, however, been left largely untouched, even though correctly accounting for the uncertainty present in typical financial applications requires the consideration of dependent data. We investigate here the theoretical and practical behaviour of two classes of extreme expectile estimators in a strictly stationary β-mixing context, containing the classes of ARMA, ARCH and GARCH models with heavy-tailed innovations that are of interest in financial applications. We put a particular emphasis on the construction of asymptotic confidence intervals adapted to the dependence framework, whose performance we contrast with that of the naive intervals obtained from the theory of independent and identically distributed data. The methods are showcased in a numerical simulation study and on real financial data.
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