Parameter Estimation for a Repairable System with Exponential Distribution

We consider a system subject to shocks that arrive according to a non-homogeneous Poisson process (NHPP). As a shock occurs, two types of failures may be happened: minor failure (type-I failure) and catastrophic failure (type-II failure). The first one is rectified by a minimal repair with probability q=1-p, whereas the second one is removed by an unplanned replacement with probability p. Sheu and Griffith (1996) considered a system that replaced at the nth type-I failure, or at any type-II failure whichever comes first. In the literature, some authors studied the optimal policy for the proposed model. Theoretical results usually suppose that the model parameters are known. However, this is not generally the case in practice. Therefore, in this paper we obtain the classical maximum likelihood and Bayes estimators of the model parameters. A likelihood ratio test statistic is also conducted to evaluate the parameters. Finally, a Mont Carlo simulation is performed to verify the behavior of the estimators and the proposed test.