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We consider a class of degradation processes that can consist of distinct phases of behavior. In particular, the degradation rates could possibly increase or decrease in a non-smooth manner at some point in time when the underlying degradation process changes phase. To model the degradation path of a given device, we use an independent-increments stochastic process with a single unobserved change-point. Furthermore, we assume that the change- point varies randomly from device-to-device. The likelihood functions for such a model are analytically intractable, so in this paper we develop an EM algorithm for this model to obtain the maximum likelihood estimators efficiently. We demonstrate the applicability of the method using two different models, and present some computational results of our implementation.