Bootstrap calibration in functional linear regression models with applications

Abstract

Our work focuses on the functional linear model given by Y = (θ,X)+ε, where Y and ε are real random variables, X is a zero-mean random variable valued in a Hilbert space (H, (·, ·), and θ ε H is the fixed model parameter. Using an initial sample {(Xi, Yi)}ni=1, a bootstrap resampling Y * i = (θ̂,Xi)+ε ̂*i , i = 1, . . . , n, is proposed, where θ̂ is a general pilot estimator, and ε̂* i is a naive or wild bootstrap error. The obtained consistency of bootstrap allows us to calibrate distributions as PX{ √ n(θ̂, x)-(θ, x) ≤ y} for a fixed x, where P X is the probability conditionally on {Xi}ni=1. Different applications illustrate the usefulness of bootstrap for testing different hypotheses related with Theta;, and a brief simulation study is also presented. © Springer-Verlag Berlin Heidelberg 2010.