Bootstrap inference in functional linear regression models

Abstract

We consider functional linear regression models (FLRMs) with functional regressor and scalar response, where the inference of the slope function is an important problem. However, even though asymptotic inference methods exist in FLRMs, these methods are limited in applicability because a wrong scaling factor is used; truncation bias in the limit is neglected; or only homoscedastic errors are assumed, which may not happen in practice. Consequently, it is necessary to develop alternative inference methods, such as bootstrap, that use the correct scaling, accommodate possible bias, and are valid even under heteroscedasticity. In this thesis, we introduce three bootstrap methods in FLRMs, namely the residual bootstrap, paired bootstrap, and wild bootstrap. Their theoretical validities are established, and their performances are numerically demonstrated. Central limit theorems for the projection are studied as well, which are fundamental results themselves and are basis to verify bootstrap validity.