Tests for the linear hypothesis in semi-functional partial linear regression models

An empirical likelihood ratio testing method is proposed, in this paper, for semi-functional partial linear regression models. Two empirical likelihood ratio statistics are employed to test the linear hypothesis of parametric components, then we demonstrate that their asymptotic null distributions are standard Chi-square distributions with the degrees of freedom being independent of the nuisance parameters. We also verify the proposed statistics follow non-central Chi-square distributions under the alternative hypothesis, and their powers are derived. Furthermore, we apply the proposed method to test the significance of parametric components. In addition, a F-test statistic is introduced. Simulations are undertaken to demonstrate the proposed methodologies and the simulation results indicate that the proposed testing methods are workable. A real example is applied for illustration.