  Extremiles provide a generalization of quantiles which are not only robust,
but also have an intrinsic link with extreme value theory. This paper
introduces an extremile regression model tailored for functional covariate
spaces. The estimation procedure turns out to be a weighted version of local
linear scalar-on-function regression, where now a double kernel approach plays
a crucial role. Asymptotic expressions for the bias and variance are
established, applicable to both decreasing bandwidth sequences and
automatically selected bandwidths. The methodology is then investigated in
detail through a simulation study. Furthermore, we highlight the applicability
of the model through the analysis of data sourced from the CH2018 Swiss climate
scenarios project, offering insights into its ability to serve as a modern tool
to quantify climate behaviour.
