Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 242567, 16 pages
Research Article

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

1Department of Mathematics, Faculty of Science, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China
2School of Electrical Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China

Received 5 June 2010; Revised 28 July 2010; Accepted 29 July 2010

Academic Editor: Ming Li

Copyright © 2010 Qihong Duan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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×m upper triangular matrices for O(nm2) times in each iteration. Finally, the algorithm is applied to two real data sets, which indicates the practicality and efficiency of our algorithm.