Estimation with Right-Censored Observations Under A Semi-Markov Model.

Area of Science:

• Statistics
• Biostatistics
• Survival Analysis

Background:

• Semi-Markov processes offer a flexible framework for multi-state event analysis, surpassing classical Markov processes in certain applications.
• Analyzing semi-Markov processes with right-censored data presents significant statistical challenges, particularly concerning identifiability and estimator consistency.
• Existing methods for constructing confidence bands for semi-Markov kernels and sojourn time distributions are often inadequate.

Purpose of the Study:

• To investigate the identifiability and consistency of semi-Markov process transition probabilities under specific right-censoring conditions.
• To develop a nonparametric inference procedure for transition probabilities using attainable values derived from censored data.
• To propose novel perturbation resampling methods for constructing confidence bands for the semi-Markov kernel and sojourn time distribution.

Main Methods:

• Derivation of the set of all attainable values for the transition probability from right-censored semi-Markov data.
• Development of a nonparametric inference procedure for transition probabilities based on the derived attainable set.
• Application of new perturbation resampling techniques, exploring various weights and transformations, for confidence band construction.

Main Results:

• Demonstrated non-identifiability and general inconsistency of existing estimators for semi-Markov transition probabilities under specific censoring scenarios.
• Established a valid nonparametric inference procedure for transition probabilities using censored data.
• Successfully constructed confidence bands for semi-Markov kernels and sojourn time distributions using novel resampling methods.

Conclusions:

• The study highlights critical limitations in current semi-Markov process analysis with censored data, particularly regarding transition probability estimation.
• The proposed nonparametric inference procedure offers a robust solution for estimating transition probabilities.
• The novel perturbation resampling methods provide reliable confidence bands, enhancing the analysis of semi-Markov processes in practical applications like cancer survivor studies.