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Title: Extremile regression Authors:  Gilles Stupfler - ENSAI - CREST (France) [presenting]
Abdelaati Daouia - Toulouse School of Economics (France)
Irene Gijbels - Katholieke Universiteit Leuven (Belgium)
Abstract: Regression extremiles define a least-squares analogue of regression quantiles. They are determined by weighted expectations rather than tail probabilities, and enjoy various closed-form expressions and interpretations. Of special interest is their intuitive meaning in terms of expected minima and expected maxima. Their use appears naturally in any decision theory where the severity of tail observations, rather than their relative frequency, is of utmost interest. In risk management, for instance, quantiles rely only on the probability of tail losses and not on their values. They also fail to fulfil the coherency axiom. Extremiles are perfectly reasonable alternatives in both of these respects. We provide the first detailed study exploring implications of the extremile terminology in a general setting of presence of covariates. We follow two paths for estimating conditional extremiles and deriving the asymptotic normality of their estimators. One is based on their characterization as the weighted average of all regression quantiles, and the other relies on local linear (least squares) check function minimization. We also extend extremile regression far into the tails of heavy-tailed distributions. Extrapolated estimators are constructed and their asymptotic extreme value theory is developed. Some applications to real data are provided.