An $Lp$ −quantile methodology for estimating extreme expectiles

Quantiles are a fundamental concept in extreme-value theory. They can be obtained from a minimization framework using an absolute error loss criterion. The companion notion of expectiles, based on squared rather than absolute error loss minimization, has recently been receiving substantial attention from the fields of actuarial science, finance and econometrics. Both of these notions can actually be embedded in a common framework of $Lp$-quantiles, whose extreme value properties have been explored very recently. However, and even though this generalized notion of quantiles has shown potential for the estimation of extreme quantiles and expectiles, it has so far not been used in the estimation of extreme value parameters of the underlying distribution of interest. In this paper, we work in a context of heavy tails, which is especially relevant to actuarial science, finance, econometrics and natural sciences, and we construct an estimator of the tail index of the underlying distribution based on extreme $Lp$-quantiles. We establish the asymptotic normality of such an estimator and in doing so, we extend very recent results on extreme expectile and $Lp$-quantile estimation. We provide a discussion of the choice of $p$ in practice, as well as a methodology for reducing the bias of our estimator. Its finite-sample performance is evaluated on simulated data and on a set of real hydrological data