Goodness-of-fit tests in the multi-state Markov model

The continuous time Markov process is commonly used to model state transitions in a multi-state system. However, it is hard to verify the Markov assumption by traditional goodness-of-fit techniques due to the multiple states. In this article, a goodness-of-fit test conditioning on a sufficient statistic is developed to test the Markov assumption. We propose a test statistic whose conditional distribution given the sufficient statistic is free of any unknown parameters. Therefore, the conditional distribution of the test statistic together with the corresponding percentiles could be obtained by Monte Carlo simulation. This conditional test outperforms two conventional goodness-of-fit techniques for the Markov assumption. Further, we consider a multi-state degradation system in which the operating conditions are available. An acceleration relation is often used to model the effects of the operating conditions on the degradation system. In this scenario, a goodness-of-fit test based on the Greenwood statistic is developed to test both the Markov assumption and the acceleration relation. The performance of the proposed goodness-of-fit tests in multi-state systems is demonstrated by extensive simulations and a numerical example.