Estimation of extreme regression risk measures

International audienceThe class of quantiles lies at the heart of extreme-value theory and is one of the basic tools in risk management. The alternative family of expectiles is based on squared rather than absolute error loss minimization. It has recently been receiving a lot of attention in actuarial science, econometrics and statistical finance. Both quantiles and expectiles can be embedded in a more general class of M-quantiles by means of Lp optimization. These generalized Lp quantiles steer an advantageous middle course between ordinary quantiles and expectiles without sacrificing their virtues too much for p between 1 and 2. Estimation methods are proposed for heavy-tailed distributions basing on extreme-value theory. Estimators of the intermediate Lp quantiles are constructed and their asymptotic normality is established, before extrapolating these estimates to the tails thus yielding extreme risk measures. A similar scheme is adopted for extreme risk measures based on conditional tail moments such as expected shortfall or conditional tail variance. We also investigate the situation where a covariate information is recorded simultaneously with the variable of interest. In this framework, we address the estimation of extreme regression risk measures thanks to nonparametric kernel methods. Some illustrations are provided both on simulated and real data.This is joint work with Abdelaati DAOUIA (Toulouse School of Economics), Jonathan ELMETHNI (Université Paris-Descartes), Laurent GARDES (Université de Strasbourg) and Gilles STUPFLER (University of Nottingham)