Composite bias-reduced L p -quantile-based estimators of extreme quantiles and expectiles

Abstract

Quantiles are a fundamental concept in extreme value theory. They can be obtained from a minimization framework using an asymmetric absolute error loss criterion. The companion notion of expectiles, based on asymmetric squared rather than asymmetric absolute error loss minimization, has received substantial attention from the fields of actuarial science, finance, and econometrics over the last decade. Quantiles and expectiles can be embedded in a common framework of $L^p$-quantiles, whose extreme value properties have been explored very recently. Although this generalized notion of quantiles has shown potential for the estimation of extreme quantiles and expectiles, available estimators remain quite difficult to use: they suffer from substantial bias, and the question of the choice of the tuning parameter $p$ remains open. In this article, we work in a context of heavy tails and construct composite bias-reduced estimators of extreme quantiles and expectiles based on $L^p$-quantiles. We provide a discussion of the data-driven choice of $p$ and of the anchor $L^p$-quantile level in practice. The proposed methodology is compared with existing approaches on simulated data and real data.