Semiparametric estimation of marginal mark distribution

In many applications, the outcome of interest is a mark such that its observation is contingent upon occurrence of an event. With incomplete follow-up data, the marginal mark distribution is, however, nonparametrically nowhere identifiable in many practical situations. To address this problem, we suggest a semiparametric model that postulates a normal copula for the association between the mark and survival time, but leaves the marginals unspecified. We show identifiability of the marginal mark distribution under this model, and propose an inference procedure. The estimated marginal distribution function is consistent and asymptotically normal, and it provides a basis for estimating summaries of the mark. Furthermore, we propose graphical model-checking methods and Kolmogorov–Smirnov-type goodness-of-fit tests. Simulation studies demonstrate that the inference procedure performs well in practical settings. The method is applied to the estimation of lifetime medical cost in a lung cancer trial.