Polynomial spline estimation of panel count data model with an unknown link function

Panel count data are frequently encountered in follow-up studies such as clinical trials, reliability researches, and insurance studies. Models about this type data usually assume the linearity form of the covariate variables on the log conditional mean function. However, the linearity assumption cannot be always guaranteed in practical applications, especially when high-dimensional covariates exist under investigation. In this paper, we propose a more flexible conditional mean regression model of panel count data with an unknown link function to describe the possible nonlinearity of the covariate effects. The partial likelihood procedure is developed to estimate the unknown link function and the regression parameters simultaneously by first approximating the unknown link function by polynomial splines, and then a two-step iterative algorithm is developed for computing implementation. Finally, the Breslow-type estimator is constructed for the baseline mean function. Asymptotic results of the proposed estimators are discussed under some regularity conditions. In addition, penalized spline estimation procedure is also introduced as an extension. Extensive numerical studies are carried out and indicate that the proposed procedure works well. Finally, two applications of bladder cancer study and skin cancer study are also presented for illustration.