An Expectation Maximization Algorithm to Model Failure Times by Continuous-Time Markov Chains

In many applications, the failure rate function may present a bathtub shape curve. In this paper, an expectation maximization algorithm is proposed to construct a suitable continuous-time Markov chain which models the failure time data by the first time reaching the absorbing state. Assume that a system is described by methods of supplementary variables, the device of stage, and so on. Given a data set, the maximum likelihood estimators of the initial distribution and the infinitesimal transition rates of the Markov chain can be obtained by our novel algorithm. Suppose that there are m transient states in the system and that there are n failure time data. The devised algorithm only needs to compute the exponential of m x m upper triangular matrices for O(nm(2)) times in each iteration. Finally, the algorithm is applied to two real data sets, which indicates the practicality and efficiency of our algorithm.