Estimating the occurrence rate for prevalent survival data in competing risks models

The problem of interest is the estimation of occurrence probabilities based on prevalent data in competing risks models. In the literature, the development of nonparametric methods has relied heavily on the independent competing risks assumption. The primary purpose of this article is to establish a statistical framework without excluding the possibility of dependence among competing risks. This is done through the use of crude hazard functions. The crude hazard functions not only are estimable regardless of whether the competing risks are independent, but also are mathematically more tractable. In this article we show that there is a one-to-one correspondence between the crude hazard functions and the occurrence probabilities. The general inversion formulas for the occurrence probabilities are presented, from which various representations can be derived under different sampling techniques. Maximum likelihood estimators are derived using these representations for nonparametric and length bias data. The maximum likelihood property and asymptotic behavior of both estimation procedures are studied. The simulation results show that the length bias estimators have smaller variances compared to the nonparametric estimators. Nevertheless, the nonparametric estimation procedure appears to be more robust to model assumptions.