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.