A Shrinkage Approach for Failure Rate Estimation of Rare Events

Abstract

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.

Integrating meta-heuristics, simulation and exact techniques for production planning of a failure-prone manufacturing system

“…In situations when no or very limited information is available about the occurrence of failures of specific production lines, statistical shrinkage estimators (Copas, 1983) may present better approaches to estimate failure rates of individual lines, as they use the information available from other production lines to determine suitable estimates. Some recent examples are presented in Xiao and Xie (2014) and Vaurio and Jänkälä (2006). The estimation of p l values via shrinkage estimators would capture a more realistic situation of this system, but is currently not implemented in our work.…”

An integrative shrinkage estimator for random-effects meta-analysis of rare binary events

“…Motivated by Xiao and Xie [ 28 ], in the k th study, we consider a shrinkage estimator of based on , denoted by , where is a fixed point in the parameter space of θ , and c is a shrinkage factor. Then our integrative shrinkage estimator for the overall treatment effect θ can be given by …”

“…To avoid this difficulty, we directly replace by in (3). This plug-in strategy was also adopted by Xiao and Xie [ 28 ], which turns out to work well in their context. We will show via simulation that it works well in the context of meta-analysis of rare binary events, too.…”

“…Based on the SA estimator, we aim to develop a shrinkage estimator with smaller MSE to improve estimation efficiency. Shrinkage methods, which shrink some “standard” estimator toward zero or any other fixed value, have been widely used in various fields [ 24 , [26] , [27] , [28] , [29] ]. Many shrinkage methods [ 7 , 8 , 29 ] were developed under a rigorous Bayesian framework via empirical Bayes (EB) approaches, which shrink a point estimate from the sample to the prior.…”

“…By contrast, the others were derived based on statistical decision theory directly (by minimizing some loss function), of which an early example is the famous James–Stein estimator [ 6 , 16 ]. A recent example is that, to estimate the failure rate of rare events from multiple heterogeneous systems, Xiao and Xie [ 28 ] developed a plug-in shrinkage estimator based on Poisson distributions, which shrinks the maximum likelihood estimator (MLE) of the failure rate toward a predetermined point in the parameter space, and so lifts degenerated zero estimates to some reasonable positive values. This estimator can be expressed as a weighted average of the MLE and a data-dependent point.…”