  Extreme quantile regression provides estimates of conditional quantiles
outside the range of the data. Classical quantile regression performs poorly in
such cases since data in the tail region are too scarce. Extreme value theory
is used for extrapolation beyond the range of observed values and estimation of
conditional extreme quantiles. Based on the peaks-over-threshold approach, the
conditional distribution above a high threshold is approximated by a
generalized Pareto distribution with covariate dependent parameters. We propose
a gradient boosting procedure to estimate a conditional generalized Pareto
distribution by minimizing its deviance. Cross-validation is used for the
choice of tuning parameters such as the number of trees and the tree depths. We
discuss diagnostic plots such as variable importance and partial dependence
plots, which help to interpret the fitted models. In simulation studies we show
that our gradient boosting procedure outperforms classical methods from
quantile regression and extreme value theory, especially for high-dimensional
predictor spaces and complex parameter response surfaces. An application to
statistical post-processing of weather forecasts with precipitation data in the
Netherlands is proposed.
