Some Asymptotic Properties of Functional Linear Regression Model withPoints of Impact

The functional linear regression model with points of impact (PoI) is a recent augmentation of the classical functional linear model with many practically important applications. It is assumed that there exists an unknown number of impact points that is, discrete observation times where the corresponding functional values possess significant influences on the response variable. In this paper, we obtain some asymptotic properties of the model that can be used for further statistical inferences about the response variable. Specifically, the rates of convergence for eigenfunctions estimates of the predictor covariance operator evaluated at the impact points estimates are derived. These are important results, because we do not have true eigenfunctions and impact points in applications and we have to use their estimates instead.