On the validity of the bootstrap hypothesis testing in functional linear regression

We consider a functional linear regression model with functional predictor and scalar response. For this model, a procedure to test the slope function based on projecting the slope function onto an arbitrary L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ L<^>2 $$\end{document} basis has been introduced in the literature. We propose its bootstrap counterpart for testing the slope function, and obtain the asymptotic null distributions of the tests statistics and the asymptotic powers of the tests. Finally, we conduct a simulation study to evaluate the accuracy of the two tests procedures. As a practical illustration, we use the Export Development Bank of Iran dataset, and test the nullity of the slope function of a model predicting total annual noncurrent balance of facilities based on current balance of facilities.