A Unified Approach to Hypothesis Testing for Functional Linear Models

A unified approach to hypothesis testing is developed for scalar-on-function, function-on-function, function-on-scalar models and particularly mixed models that contain both functional and scalar predictors. In contrast with most existing methods that rest on the large-sample distributions of test statistics, the proposed method leverages the technique of bootstrapping max statistics and exploits the variance decay property that is an inherent feature of functional data, to improve the empirical power of tests especially when the sample size is limited or the signal is relatively weak. Theoretical guarantees on the validity and consistency of the proposed test are provided uniformly for a class of test statistics.