Bootstrap Calibration in Functional Linear Regression Models with Applications

Abstract

Our work focuses on the functional linear model given by $Y=\langle\theta,X\rangle+\epsilon,$ where Y and ε are real random variables, X is a zero-mean random variable valued in a Hilbert space $(\mathcal{H},\langle\cdot,\cdot\rangle)$, and $\theta\in\mathcal{H}$ is the fixed model parameter. Using an initial sample $\{(X_i,Y_i)\}_{i=1}^n$, a bootstrap resampling $Y_i^{*}=\langle\hat{\theta},X_i\rangle+\hat{\epsilon}_i^{*}$, $i=1,\ldots,n$, is proposed, where $\hat{\theta}$ is a general pilot estimator, and $\hat{\epsilon}_i^{*}$ is a naive or wild bootstrap error. The obtained consistency of bootstrap allows us to calibrate distributions as $P_X\{\sqrt{n}(\langle\hat{\theta},x\rangle-\langle\theta,x\rangle)\leq y\}$ for a fixed x, where P _X is the probability conditionally on $\{X_i\}_{i=1}^n$. Different applications illustrate the usefulness of bootstrap for testing different hypotheses related with θ, and a brief simulation study is also presented.