This paper proposes an improved expectation–maximization (EM) algorithm for phase-type (PH) distributions with grouped and truncated data. Olsson (1996) derived an EM algorithm for PH distributions under censored data, and the similar technique can be utilized to the PH fitting even under grouped and truncated data. However, it should be noted that Olsson's algorithm has a drawback in terms of computation speed. Because the time complexity of the algorithm is a cube of number of phases, it does not work well in the case where the number of phases is large. This paper proposes an improvement of the EM algorithm under grouped and truncated observations. By applying a uniformization-based technique for continuous-time Markov chains, it is shown that the time complexity of our algorithm can be reduced to the square of number of phases. In particular, when we consider the PH fitting using a canonical form of PH distributions, the time complexity is linear in the number of phases. Copyright © 2012 John Wiley & Sons, Ltd.
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