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 illustrate the method's applicability with an analysis of the Berkeley Growth data, showcasing its performance in a real-world functional data setting.

翻译：极值分位数提供了一种对分位数的推广，它不仅具有稳健性，而且与极值理论存在内在联系。本文提出了一种针对函数型协变量空间设计的极值分位数回归模型。其估计过程可转化为加权版本的局部线性标量对函数回归，其中双核方法起着关键作用。我们建立了偏差和方差的渐近表达式，适用于递减带宽序列和自动选择的带宽。随后通过模拟研究对该方法进行了详细探讨。此外，我们通过对伯克利生长数据的分析展示了该方法在实际函数型数据场景中的适用性及其表现。