Asymptotic properties of the GMLE with case 2 interval-censored data

In case 2 interval censoring the random survival time X of interest is not directly observable, but only known to have occurred before Y, between Y and Z, or after Z, where (Y, Z) is a pair of observable inspection times such that Y < Z. We consider the large sample properties of the generalized maximum likelihood estimator (GMLE) of the distribution function of X with case 2 interval-censored data in which the inspection times are discrete random variables. We prove the strong consistency of the GMLE at the support points of the inspection times and establish its asymptotic normality in the case of only finite many support points.