Every mechanical system is naturally subjected to some kind of wear process that, at some point, will cause failure in the system if no monitoring or treatment process is applied. Since failures often lead to high economical costs, it is essential both to predict and to avoid them. To achieve this, a monitoring system of the wear level should be implemented to decrease the risk of failure. In this work, we develop a multiscale indirect inference methodology for state-dependent Markovian pure jump processes that allows us to model the evolution of the wear level, and to identify when the system reaches some critical level that triggers a maintenance response.

Since the likelihood function of a discretely observed pure jump process does not have an expression that is simple enough for standard non-sampling optimization methods, we approximate this likelihood by expressions from upscaled models of the data.

We use the Master Equation to assess the goodness-of-fit and to compute the distribution of the hitting time to the critical level.