Estimation for the distribution function of one- and two-dimensional censored variables or sojourn times of Markov renewal processes

Self-consistency equations are established for the distribution functions of right and left-censored one- and two-dimensional variables and sojourn times of a Markov renewal process. They have a unique solution that equals the product-limit estimator if a hazard function may be defined. Under right-censoring, the results presented here provide new formulations of known estimators. If the left- and right-censoring times are dependent, no estimators were available and simple algorithms are defined. All the estimators are,root n-consistent and converge weakly to centered Gaussian processes.