Estimation for the Distribution Function of One- and Two-Dimensional Censored Variables or Sojourn Times of Markov Renewal Processes

Abstract

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 -consistent and converge weakly to centered Gaussian processes.

Keywords:

• Bivariate distribution function
• Double censoring
• Functional Z-estimators
• Markov renewal process
• Self-consistency equations
• Weak convergence

Mathematics Subject Classification:

• Primary 62M05
• Secondary 62H11, 62G20