In this paper, we discuss the theoretical validity of the observed partial likelihood (OPL) constructed in a Coxtype model under incomplete data with two class possibilities, such as missing binary covariates, a cure-mixture model or doubly censored data. A main result is establishing the asymptotic convergence of the OPL. To reach this result, as it is difficult to apply some standard tools in the survival analysis, we develop tools for weak convergence based on partial-sum processes. The result of the asymptotic convergence shown here indicates that a suitable order of the number of Monte Carlo trials is less than the square of the sample size. In addition, using numerical examples, we investigate how the asymptotic properties discussed here behave in a finite sample.