Nonparametric inference for Markov processes with missing absorbing state

Abstract

This study examines nonparametric estimations of a transition proba- bility matrix of a nonhomogeneous Markov process with a nite state space and a partially observed absorbing state. We impose a missing-at-random assumption and propose a computationally e cient 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- t 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 eld. We also provide a methodology for constructing simultaneous con dence bands. Simulation studies show that the NPMPLE works well with small sample sizes and that it is robust against some degree of misspec- i cation 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.