Bootstrap independence test for functional linear models

Functional data have been the subject of many research works over the last years. Functional regression is one of the most discussed issues. Specifically, significant advances have been made for functional linear regression models with scalar response. Let $(\mathcal{H},<\cdot ,\cdot >)$ be a separable Hilbert space. We focus on the model $Y=<\mathrm{\Theta },X>+b+\epsilon$, where $Y$ and $\epsilon$ are real random variables, $X$ is an $\mathcal{H}$-valued random element, and the model parameters $b$ and $\mathrm{\Theta }$ are in $\mathbb{R}$ and $\mathcal{H}$, respectively. Furthermore, the error satisfies that $E(\epsilon \mathrm{\mid }X)=0$ and $E({\epsilon }^2\mathrm{\mid }X)={\sigma }^2<\mathrm{\infty }$. A consistent bootstrap method to calibrate the distribution of statistics for testing $H_0: \Theta=0$versus$H_1: \Theta\neq 0$ is developed. The asymptotic theory, as well as a simulation study and a real data application illustrating the usefulness of our proposed bootstrap in practice, is presented.