Testing the significance of index parameters in varying-coefficient single-index models

The varying-coefficient single-index models (VCSIMs) form a class of very flexible and general dimension reduction models, which contain many important regression models such as partially linear models, pure single-index models, partially linear single-index models, varying-coefficient models and so on as special examples. However, the testing problems of the index parameter of the VCSIM have not been very well developed, due partially to the complexity of the models. To this end, based on the estimators obtained by the local linear method and the backfitting technique, we propose the generalized F-type test method to deal with the testing problems of the index parameters of the VCSIM. It is shown that under the null hypothesis the proposed test statistic follows asymptotically a χ2-distribution with the scale constant and the degrees of freedom being independent of the nuisance parameters or functions, which is called the Wilks phenomenon. Simulated data and real data examples are used to illustrate our proposed methodology.