Expectiles dene a least squares analogue of quantiles. They are determined by tail expectations rather than tail probabilities. For this reason and many other theoretical and practical merits, expectiles have recently received a lot of attention, especially in actuarial and nancial risk management. Their estimation, however, typically requires to consider non-explicit asymmetric least squares estimates rather than the traditional order statistics used for quantile estimation. This makes the study of the tail expectile process a lot harder than that of the standard tail quantile process. Under the challenging model of heavy-tailed distributions, we derive joint weighted Gaussian approximations of the tail empirical expectile and quantile processes. We then use this powerful result to introduce and study new estimators of extreme expectiles and the standard quantile-based expected shortfall, as well as a novel expectile-based form of expected shortfall. Our estimators are built on general weighted combinations of both top order statistics and asymmetric least squares estimates. Some numerical simulations and applications to actuarial and nancial data are provided.
Asymmetric least squares; Coherent risk measures; Expected shortfall; Expectile; Extrapolation; Extremes; Heavy tails; Tail index;