  Extremile (Daouia, Gijbels and Stupfler,2019) is a novel and coherent measure
of risk, determined by weighted expectations rather than tail probabilities. It
finds application in risk management, and, in contrast to quantiles, it
fulfills the axioms of consistency, taking into account the severity of tail
losses. However, existing studies (Daouia, Gijbels and Stupfler,2019,2022) on
extremile involve unknown distribution functions, making it challenging to
obtain a root n-consistent estimator for unknown parameters in linear extremile
regression. This article introduces a new definition of linear extremile
regression and its estimation method, where the estimator is root n-consistent.
Additionally, while the analysis of unlabeled data for extremes presents a
significant challenge and is currently a topic of great interest in machine
learning for various classification problems, we have developed a
semi-supervised framework for the proposed extremile regression using unlabeled
data. This framework can also enhance estimation accuracy under model
misspecification. Both simulations and real data analyses have been conducted
to illustrate the finite sample performance of the proposed methods.
