Parameter estimation in progressive multistate models

Abstract

Multistate models are widely used to analyze systems or entities that transition between different states over time. These models have applications in various fields, including survival analysis, reliability engineering, and financial mathematics. This thesis will focus on parameter estimation for time-homogeneous progressive Markov chains, where transitions are limited to higher-numbered states, and the final state is absorbing. The thesis can be divided into two parts, the first part focusing on normal multistate models and the other part focusing on hidden multistate models. In the first part, we study progressive, time-homogeneous multistate Markov chains. During this part, maximum likelihood estimators (MLEs) for transition probabilities are derived for uncensored, fixed-censored, and random-censored scenarios. Consistency proofs for these estimators are provided using theoretical tools such as the Strong Law of Large Numbers and the Continuous Mapping Theorem. The second part focuses on hidden multistate models, specifically the twostate hidden Markov chains. In this case, the underlying states are not fully observable: In each observation, each position of the chain has a constant probability of being observed. We study the block structure of the observations and derive statistical properties of block lengths under different censoring mechanisms. Here we only have one transition probability to estimate, moment-based estimators and maximum likelihood estimators are developed for the transition probability, and their consistency is validated through simulations. Simulation experiments are also conducted for both parts to evaluate the performance of the derived estimators under various parameter configurations and censoring mechanisms. The results confirm the consistency of all proposed estimators.