Inference for function-on-function regression: central limit theorem and residual bootstrap

We investigate asymptotic inference in a linear regression model where both response and regressors are functions, using an estimator based on functional principal components analysis. Although this approach is widely used in functional data analysis, there remains significant room for developing its asymptotic properties for function-on-function regression. Our study targets the mean response at a new regressor with two primary aims. First, we refine the existing central limit theorem by relaxing certain technical conditions, which include generalizing the scaling factor, resulting in incorporating a broader class of random functions beyond those having scores with independence or finite higher moments. Second, we introduce a residual bootstrap method that enhances the calibration of various confidence sets for quantities related to mean response, while its consistency is rigorously verified. Numerical studies compare the finite sample performance of both asymptotic and bootstrap approaches, demonstrating higher accuracy of the latter. To illustrate bootstrap inference for mean response, we apply it to the Canadian weather dataset.