Regularized partially functional quantile regression

We propose a regularized partially functional quantile regression model where the response variable is scalar while the explanatory variables involve both infinite-dimensional predictor processes viewed as functional data, and high-dimensional scalar covariates. Despite extensive work focusing on functional linear models, little effort has been devoted to the development of robust methodologies that tackle the scenarios of non-normal errors. This motivates our proposal of functional quantile regression that seeks an alternative and robust solution to least squares type procedures within the partially functional regression framework. We focus on estimating and selecting the important variables in the high-dimensional covariates, which is complicated by the infinite-dimensional functional predictor. We establish the asymptotic properties of the resulting shrinkage estimator, and empirical illustrations are given by simulation and an application to a brain imaging dataset.