Functional Linear Quantile Regression on a Two-dimensional Domain

This article considers the functional linear quantile regression which models the conditional quantile of a scalar response given a functional predictor over a two-dimensional domain. We propose an estimator for the slope function by minimizing the penalized empirical check loss function. Under the framework of reproducing kernel Hilbert space, the minimax rate of convergence for the regularized estimator is established. Using the theory of interpolation spaces on a two- or multi-dimensional domain, we develop a novel result on simultaneous diagonalization of the reproducing and covariance kernels, revealing the interaction of the two kernels in determining the optimal convergence rate of the estimator. Sufficient conditions are provided to show that our analysis applies to many situations, for example, when the covariance kernel is from the Mat\'ern class, and the slope function belongs to a Sobolev space. We implement the interior point method to compute the regularized estimator and illustrate the proposed method by applying it to the hippocampus surface data in the ADNI study.