Precision of power-law NHPP estimates for multiple systems with known failure rate scaling

The power-law non-homogeneous Poisson process, also called the Crow-AMSAA model, is often used to model the failure rate of repairable systems. In standard applications it is assumed that the recurrence rate is the same for all systems that are observed. The estimation of the model parameters on the basis of past failure data is typically performed using maximum likelihood. If the operational period over which failures are observed differs for each system, the Fisher information matrix is numerically inverted to quantify the precision of the parameter estimates. In this paper, the extended case is considered where the recurrence rate between the different systems may vary with known scaling factors and it is shown that the standard error of the parameter estimates can be quantified using analytical formulae. The scaling factors allow to apply the model to a wider range of problems. The analytical solution for the standard error simplifies the application and allows to better understand how the precision of the model varies with the extent of available data. The good performance and the practical use of the method is illustrated in an example.