Bootstrap Calibration in Functional Linear Regression Models with Applications

摘要

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 ∈ H is the fixed model parameter. Using an initial sample {(X_i,Y_i)}_(i=1)~n, a bootstrap resampling Y_i~*=<θ,X_i>+∈_i~*, i=1,...,n, is proposed, where θ is a general pilot estimator, and ∈_i~* is a naive or wild bootstrap error. The obtained consistency of bootstrap allows us to calibrate distributions as P_X{n(1/2)(<θ,x>-<θ,x>)≤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.