Hypothesis testing for points of impact in functional linear regression

Recently, there has been increased interest in issues related to functional linear regression models with points of impact. While the estimation of model parameters with a scalar response has been considered in past studies, there has been no attention on the hypothesis testing for these impact points. To test this hypothesis, one needs to determine the asymptotic distribution of the impact points coefficients estimators. In recent literature, the asymptotic distribution has been pointed out in a special case, but the proof has not been provided. Taking into account the necessary conditions, this study establishes the asymptotic distribution for the estimators of impact points coefficients in a general setting. It also offers a method to test the significance of these impact points. To validate the proposed test's asymptotic properties, a simulation study is conducted to assess its performance under various parameter settings. Furthermore, the study analyzes Iranian weather data collected from January 1st to 31st, 2023.