Abstract: In this study, we examine two-sample functional linear regressions with a scaling transformation of the regression functions. We estimate the intercept, slope function, and scalar parameter using a functional principal component analysis. We also establish the rate of convergence of the estimator of the slope function, which is shown to be optimal in a minimax sense under certain smoothness assumptions. In addition, we investigate the semiparametric efficiency of the estimation of the scalar parameter and the hypothesis tests. Then, we extend the proposed method to include sparsely and irregularly sampled functional data and establish the consistency of the estimators of the scalar parameter and the slope function. We evaluate the numerical performance of the proposed methods through simulation studies and illustrate their utility via an analysis of an AIDS data set.

Key words and phrases: Functional linear regression, functional principal component analysis, hypothesis testing, minimax rate of convergence, semiparametric comparison, semiparametric efficiency.