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.

中文翻译：

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具有缺失吸收状态的马尔可夫过程的非参数推理

本文涉及具有有限状态空间和部分观测吸收状态的非齐次马尔可夫过程的转移概率矩阵的非参数估计问题。我们强加了一个随机缺失的假设，并提出了一个计算效率高的非参数最大伪似然估计器（NPMPLE）。估计器依赖于参数模型，该模型用于估计每个吸收状态的概率，以潜在地基于辅助数据。对于后一种模型，我们提出了基于残差过程的正式拟合优度测试。使用现代经验过程理论，我们表明估计量是一致一致的，并且弱收敛到严格的均值零高斯随机场。我们还提供了同时构建置信带的方法。模拟研究表明，NPMPLE 在样本量较小的情况下运行良好，并且对于缺失吸收状态的参数模型的某种程度的错误指定具有鲁棒性。该方法使用来自撒哈拉以南非洲的 HIV 数据来说明，以估计死亡和脱离 HIV 护理的过渡概率。