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Inference on Large-scale Partially Functional Linear Model with Heterogeneous Errors

We investigate a partially functional linear model by focusing on the heterogenous error scenario that 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 rigorous inferential procedure for hypothesis testing on an arbitrary subset of both regression functions and scalar coefficients. To be specific, we devise a confidence region for post-regularization inference via a pseudo score function that is not decorrelated due to heterogenous errors. The proposed test does not require the estimation consistency of functional part and is shown uniformly convergent to the prescribed significance, and its finite sample performance is illustrated via simulation studies and an application to a motivating functional magnetic resonance imaging brain image data.