Abstract: In this work, we extend the classical generalized functional linear model to a large-scale generalized functional linear model to handle a variety of complex situations where the response (possibly discrete) can be nonlinearly linked to an ultra-high number of functional predictors. Unlike most existing requirements on functional data, we don't need to impose any conditions regarding eigenvalue-decay or square-integrability on those functional predictors, resulting in a more flexible but challenging model framework. Based on a penalized model estimator, we develop a general inferential method to assess the significance of an arbitrary group of regression curves. Concretely, a pseudo score function is adopted to construct the associated confidence region for the regression curves of interest. Notably, the proposed test is justified uniformly convergent to nominal level, without any demand on estimation consistency of the regression curves. Finally, numerical studies are carried out to show the empirical performance of the proposed test.

Key words and phrases: Eigenvalue-decay-free, estimation-consistency-relaxed, high dimensions, multiplier bootstrap, square-integrable-free.