In this communication, we propose a new approach, called Extreme-PLS, for dimension reduction in regression and adapted to distribution tails. The goal is to find linear combinations of predictors that best explain the extreme values of the response variable by maximizing the associated covariance. This adaptation of the PLS estimator to the extreme-value framework is achieved in the context of a non-linear inverse regression model. In practice, it allows to quantify the effect of the covariates on the extreme values of the response variable in a simple and interpretable way. Moreover, it should yield improved results for most estimators dealing with conditional extreme values thanks to the dimension reduction achieved in the projection step. From the theoretical point of view, the asymptotic normality of the Extreme-PLS estimator is established under a heavy tail assumption but without recourse to linearity nor independence assumptions.
We provide also a Bayesian extension to the Extreme-PLS method to address data scarcity problems in distribution tails. This approach allows to identify the direction of dimension reduction by introducing a prior information on it. It provides a Bayesian framework for computing the posterior distribution of the direction, where the likelihood function is obtained from a von Mises-Fisher distribution adapted to hyperballs. Finally, the performance of both approaches is evaluated on simulated data, and an application on French farm income data is provided as an illustration.