A spline-based semiparametric sieve likelihood method for over-dispersed panel count data

In this article we study a Gamma-Frailty inhomogeneous Poisson process model for analysing over-dispersed panel count data. A cubic B-spline function is used to approximate the logarithm of the baseline mean function in the semiparametric proportional mean model. The regression parameters and spline coefficients are jointly estimated by maximizing a spline-based sieve pseudo-likelihood and by replacing the nuisance over-dispersion parameter with its moment estimate. The asymptotic properties of the proposed maximum pseudo likelihood estimator, including its consistency, convergence rate and the asymptotic normality of the estimated regression parameters, are thoroughly studied using modern empirical process theory. A spline-based least-squares standard error estimator is developed to facilitate robust inference for the regression parameters. Simulation studies are conducted to investigate finite sample performance of the proposed method and robustness of the Gamma-Frailty inhomogeneous Poisson process model. Finally, for illustration, the method is used to analyse data from an observational study of sexually transmitted infection (STI) in young women.