Uniqueness of Maximum Likelihood Estimators for a Backup System in a Condition-Based Maintenance

A parameter estimation problem for a backup system in a condition-based maintenance is considered. We model a backup system by a hidden, three-state continuous time Markov process. Data are obtained through condition monitoring at discrete time points. Maximum likelihood estimates of the model parameters are obtained using the EM algorithm. We establish conditions under which there is no more than one limitation in the parameter space for any sequence derived by the EM algorithm. 1. Introduction Suppose a backup system is represented by a continuous time homogeneous Markov chain with a state space . States 0, 1, and 2 are the healthy state, unhealthy state, and the failure state, respectively. Assume that the system is in a healthy state at time 0, and the transition rate matrix is given by where for are unknown. Here is a known extreme edge of the parameters. Suppose the system is observed at time points , where is a benchmark interval. While the system is failed at an inspection, a new system replaces it. Let two processes , and let be a record of the system. Here if the system is failed during , and otherwise. And . As a path of is a stepped right-continuous function, when a replacement occurs at time . Moreover, we set for convenience. The process represents the replacement of the system, and the process represents observable information of the system collected through condition monitoring. The maximum likelihood estimates (MLE) of the model parameters for such models have been studied by [1, 2]. As the stochastic processes, are not Markov processes and the sample path of is not observable, the likelihood function of incomplete data is complex. Hence, it is difficult to obtain directly the MLE , where and . Here is a prearranged constant. Both [1, 2] suggest the EM algorithm (see e.g., [3, 4]). Let be initial values of the unknown parameters. The EM algorithm works as follows. The step. For , compute the following pseudo-likelihood function: Here is the complete data set of the process . The forms of the complete data set may be different for different purpose. For example, the forms are different in [1, 2]. The step. Choose . Here The and steps are repeated. According to the theory of EM algorithms (Theorem？？1？in [3]), for any given initial value and . It is clear that if an MLE in is one of these fixed-points when it exists. In this paper, we consider the uniqueness of the MLE . As the likelihood function of incomplete data is complex, we do not follow the classical method by which the uniqueness of a MLE is demonstrated by establishing the global