Checking the adequacy of functional linear quantile regression model

The functional linear quantile regression model is widely used to characterize the relationship between a scalar response and a functional covariate. Most existing research results are based on a correct assumption that the response is related to the functional predictor through a linear model for given quantile levels. This paper focuses on investigating the adequacy check of the functional linear quantile regression model. We propose a nonparametric U-process test statistic based on the functional principal component analysis. It is proved that the test statistic follows a normal distribution asymptotically under the null hypothesis and diverges to infinity for any misspecified models. Therefore, the test is consistent against any fixed alternative. Moreover, it is shown that the test has asymptotic power one for the local alternative hypothetical models converging to the null hypothesis at the rates n−12. The finite sample properties of the test statistic are illustrated through extensive simulation studies. A real data set of 24 hourly measurements of ozone levels in Sacramento, California is analyzed by the proposed test.