A Shrinkage Approach for Failure Rate Estimation of Rare Events

Systems have become more and more reliable due to technological advancement. For highly reliable systems, there are usually very few or even no failures during the testing and operation. On the other hand, given a short operating or testing time, the failure of the system is also rare even the failure rate is relatively high. The classical maximum likelihood estimation approach results in degenerated estimates zero when no failure occurs and hence meaningless. To overcome this problem, we investigate a shrinkage approach for the estimation of the failure rate of rare events from multiple heterogeneous systems provided that some of them have failures. The shrinkage estimator shrinks the MLE toward a predetermined data-dependent point in the parameter space and could be expressed as a weighted average of MLE and the data-dependent point. Examples are shown how the procedure can be implemented. Simulation studies show that our approach performs better than the other approaches in most cases.