Abstract: In this study, we examine inferences (in particular, hypothesis tests) for generalized partial functional linear models. A Bahadur representation for both functional and scalar estimators is developed based on the reproducing kernel Hilbert space (RKHS). We establish the asymptotic independence between the scalar estimators and the estimator of the functional part. A penalized likelihood ratio test is proposed to detect the significant effects of the functional and scalar covariates on the scalar outcome, either simultaneously or separately. The asymptotic normality of the test statistic is established under the null hypothesis. Simulation studies provide numerical support for the asymptotic properties. Lastly, data on air pollution are used to demonstrate our method.

Key words and phrases: Bahadur representation, hypothesis testing, penalized likelihood ratio test, reproducing kernel Hilbert space.