| In this dissertation, we are concerned with estimation problems for counting processes with incomplete data. We focus mainly on methods for one general type of incomplete data, panel count data. We study nonparametric and semiparametric methods for panel count data with and without covariates.;In the first part of the dissertation, we consider the situation of panel count data without covariates, and study some nonparametric methods for estimating the mean function of a counting process. Two approaches are investigated: (1) We assume that the underlying counting process is a nonhomogeneous Poisson process, but ignore the dependence of panel counts within a subject, which leads to a so-called pseudo likelihood function. Maximizing the pseudo log-likelihood function leads to a one-step algorithm for computing the nonparametric maximum pseudo likelihood estimator (NPMPLE). (2) We again assume a nonhomogeneous Poisson process for the underlying counting process, and derive the full likelihood function for the observed data based on this assumption. The full nonparametric maximum likelihood estimator (NPMLE) of the mean function is then computed by the modified iterative convex minorant algorithm (MDICM). Under some mild hypotheses, we establish consistency theorems for both NPMPLE, and NPMLE, as well as an asymptotic distribution theorem for the NPMPLE. We establish a very strong robustness property of these estimators: they converge to the true mean function, regardless of the true underlying counting process for the data. Two simulation studies (one data set is generated from a Poisson process, and the other is not) support this robustness, but indicate that the NPMLE is more efficient than the NPMPLE.;In the second part of the dissertation, we consider the situation of panel count data with time-independent covariates, and study a semiparametric method for estimating both the baseline mean function and regression parameters with concentration on the regression parameters. For this type of data, we assume that given covariates Z, the underlying counting process is a nonhomogeneous Poisson process with conditional mean function given by... |
