Bootstrap Calibration in Functional Linear Regression Models with Applications

Our work focuses on the functional linear model given by Y = <theta,X > +is an element of, where Y and is an element of are real random variables, X is a zero-mean random variable valued in a Hilbert space (H, <center dot,center dot >), and theta is an element of H is the fixed model parameter. Using an initial sample {(X-i,Y-i)}(i=1)(n), a bootstrap resampling Y-i(*) = <theta,X-i > + is an element of(i), i = 1,..., n, is proposed, where theta is a general pilot estimator, and is an element of(*)(i) is a naive or wild bootstrap error. The obtained consistency of bootstrap allows us to calibrate distributions as P-X {[GRAPHIC](<theta,x > - <theta,x >) <= y} for a fixed x, where Px is the probability conditionally on {X-i}(i=1)(n). Different applications illustrate the usefulness of bootstrap for testing different hypotheses related with theta, and a brief simulation study is also presented.