Abstract: We investigate a partially functional linear model by focusing on the heterogenous error scenario in which the scalar response is associated with an ultra-large number of both functional predictors and scalar covariates. Moreover, the model does not require the standard condition on eigenvalue decay for functional predictors, leading to a more challenging and general framework. The target is to establish a rigorous inferential procedure for hypothesis testing on an arbitrary subset of both regression functions and scalar coefficients. Specifically, we devise a confidence region for post-regularization inference using a pseudo score function that is not decorrelated owing to the heterogenous errors. The proposed test does not require estimation consistency of the functional part, and is shown to be uniformly convergent to the prescribed significance. We investigate the finite-sample performance of the proposed model using simulation studies and an application to functional magnetic resonance imaging brain image data.

Key words and phrases: Eigenvalue-decay-relaxation, high dimensions, multiplier bootstrap.