TAIL EXPECTILE ESTIMATION IN THE SEMIPARAMETRIC GENERALIZED PARETO MODEL

Expectiles have received increasing attention as a market risk measure that is both coherent and elicitable. They are defined as a least squares analog to quantiles. Their estimation from heavy-tailed loss data in an extreme value framework is an important and difficult problem, especially when the target tail expectile is beyond the range of the data. This problem has been studied only fairly recently, using solely the Weissman extrapolation method. The competing generalized Pareto approach, which makes efficient use of tail observations by incorporating the location, scale, and shape extreme value parameters into the estimation procedure, has been left untouched, with a corresponding theory completely lacking. In this article, we challenge the dominance of the Weissman device by presenting and developing the theory of two classes of semiparametric Generalized Pareto estimators: the first class relies on direct asymmetric least squares estimation, while the second is based on extreme quantile estimation. Our estimators are found to outperform the best-known Weissman-type estimators for real-valued profit–loss distributions, while staying competitive for nonnegative loss variables. A forecast comparison exercise is also conducted on various sets of financial returns, showing the superiority of the generalized Pareto approach.