Nonparametric inference for markov processes with missing absorbing state

This study examines nonparametric estimations of a transition probability matrix of a nonhomogeneous Markov process with a finite state space and a partially observed absorbing state. We impose a missing-at-random assumption and propose a computationally efficient nonparametric maximum pseudolikelihood estimator (NPMPLE). The estimator depends on a parametric model that is used to estimate the probability of each absorbing state for the missing observations based, potentially, on auxiliary data. For the latter model, we propose a formal goodness-of-fit test based on a residual process. Using modern empirical process theory, we show that the estimator is uniformly consistent and converges weakly to a tight mean-zero Gaussian random field. We also provide a methodology for constructing simultaneous confidence bands. Simulation studies show that the NPMPLE works well with small sample sizes and that it is robust against some degree of misspecification of the parametric model for the missing absorbing states. The method is illustrated using HIV data from sub-Saharan Africa to estimate the transition probabilities of death and disengagement from HIV care.