Statistical inference for the index parameter in single-index models

In this paper, we are concerned with statistical inference for the index parameter α0 in the single-index model Y=g(α0TX)+ϵ. Based on the estimates obtained by the local linear method, we extend the generalized likelihood ratio test to the single-index model. We investigate the asymptotic behaviour of the proposed test and demonstrate that its limiting null distribution follows a χ2χ2-distribution, with the scale constant and the number of degrees of freedom being independent of nuisance parameters or functions, which is called the Wilks phenomenon. A simulated example is used to illustrate the performance of the testing approach.