Inference for a Failure Counting Process Partially Observed

We consider a system defined as a collection of two types of components. The number of failures of each component is described as a stochastic process, with one of the processes depending on the other. None of the processes is observed directly. The only available information is the number of type 1 components at risk in the system. Because of this missing data situation, different algorithms relying on an Expectation Maximization (EM) strategy are proposed to obtain the MLE of the intensity parameters for both processes so we can assess the reliability of type 1 and type 2 components. To overcome the computational limits of EM, a Monte Carlo EM (MCEM) algorithm using a Metropolis procedure is presented. Stochastic EM (SEM) algorithms including a Bayesian approach are also described. The methods are applied to simulated data to demonstrate their efficiency.