Testing for the parametric parts in a single-index varying-coefficient model

Single-index varying-coefficient models (SIVCMs) are very useful in multivariate nonparametric regression. However, there has less attention focused on inferences of the SIVCMs. Using the local linear method, we propose estimates of the unknowns in the SIVCMs. In this article, our main purpose is to examine whether the generalized likelihood ratio (GLR) tests are applicable to the testing problem for the index parameter in the SIVCMs. Under the null hypothesis our proposed GLR statistic follows the chi-squared distribution asymptotically with scale constant and degree of freedom independent of the nuisance parameters or functions, which is called as Wilks' phenomenon (see Fan et al., 2001). A simulation study is conducted to illustrate the proposed methodology.