Profile-kernel likelihood inference with diverging number of parameters

The generalized varying coefficient partially linear model with a growing number of predictors arises in many contemporary scientific endeavor. In this paper we set foot on both theoretical and practical sides of profile likelihood estimation and inference. When the number of parameters grows with sample size, the existence and asymptotic normality of the profile likelihood estimator are established under some regularity conditions. Profile likelihood ratio inference for the growing number of parameters is proposed and Wilk's phenomenon is demonstrated. A new algorithm, called the accelerated profile-kernel algorithm, for computing profile-kernel estimator is proposed and investigated. Simulation studies show that the resulting estimates are as efficient as the fully iterative profile-kernel estimates. For moderate sample sizes, our proposed procedure saves much computational time over the fully iterative profile-kernel one and gives stabler estimates. A set of real data is analyzed using our proposed algorithm.