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Title: Extremile regression Authors:  Irene Gijbels - Katholieke Universiteit Leuven (Belgium)
Gilles Stupfler - ENSAI - CREST (France)
Abdelaati Daouia - Fondation Jean-Jacques Laffont (France) [presenting]
Abstract: Regression extremiles define a least squares analogue of regression quantiles. They are determined by weighted expectations rather than tail probabilities. Of special interest is their intuitive meaning in terms of expected minima and maxima. Their use appears naturally in risk management where, in contrast to quantiles, they fulfill the coherency axiom and take the severity of tail losses into account. In addition, they are co-monotonically additive and belong to both families of spectral risk measures and concave distortion risk measures. The aim is to provide the first detailed study exploring implications of the extremile terminology in a general setting of presence of covariates. We rely on local linear (least squares) check function minimization for estimating conditional extremiles and deriving the asymptotic normality of their estimators. We also extend extremile regression far into the tails of heavy-tailed distributions. Extrapolated estimators are constructed and their asymptotic theory is developed. Some applications to real data are provided.