Nonparametric Inference for Markov Processes with Missing Absorbing State

This paper deals with the issue of nonparametric estimation of the transition probability matrix of a non-homogeneous Markov process with finite state space and 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 methodology for simultaneous confidence band construction. 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.