Functional estimation of extreme conditional expectiles

Quantiles and expectiles can be interpreted as solutions of convex minimization problems. Unlike quantiles, expectiles are determined by tail expectations rather than tail probabilities, and define a coherent risk measure. For these reasons, among others, they have recently been the subject of renewed attention in actuarial and financial risk management. The challenging problem of estimating extreme expectiles, whose order converges to one as the sample size increases, is considered given a functional covariate. A functional kernel estimator of extreme conditional expectiles is constructed by writing expectiles as quantiles of a different distribution. The asymptotic properties of the estimators are studied in the context of conditional heavy-tailed distributions. Different ways of estimating the functional tail index, as a way to extrapolate the estimates to the very far conditional tails, are examined. A numerical illustration of the finite-sample performance of the estimators is provided on simulated and real datasets.