A Convergence Rate in Functional Linear Regression with Points of Impact

Authors
Alireza Shirvani 1
mohammad Hosseini-Nasab 2
Omid Khademnoe 3
1 Velayat University
2 Shahid Beheshti University
3 University of Zanjan
Abstract
The classical functional linear regression model with a scalar response has been widely used. However, the local effects of the predictor function on the response cannot be evaluated with this model, as it considers only a weighted average of the entire predictor trajectory in relation to the response. To ad-dress this issue, we propose a generalization of the classical functional linear regression model by adding an unknown number of points of impact, where the predictor values at these points have a significant effect on the response. The points of impact and their coefficients, the slope function, the eigenvalues, and the eigenfunctions of the covariance operator of the predictor are unknown and need to be estimated. In this paper, we derive the convergence rate of a quantity constructed based on these unknown parameters. This result can be used in future statistical inferences for this model, including the calculation of confidence in-tervals and bootstrap confidence intervals. Additionally, a simulation study has been conducted to evaluate the obtained results.
Keywords
Functional linear regression
Point of impact
Convergence rate
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