## An expectation maximization algorithm to model failure times by continuous-time Markov chains.(English)Zbl 1203.90055

Summary: 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 $$m\times m$$ 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.

### MSC:

 90B25 Reliability, availability, maintenance, inspection in operations research 65C60 Computational problems in statistics (MSC2010) 60J22 Computational methods in Markov chains

### Keywords:

supplementary variables; maximum likelihood estimators
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### References:

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