INTERVAL ESTIMATION FROM CENSORED AND MASKED SYSTEM-FAILURE DATA

In a competing risk (or multiple failure mode) framework, masking occurs when a failure is observed without full knowledge of the cause of system failure. Exact confidence interval (CI) estimation from such data is usually not tractable, and therefore large-sample approximate CI estimation is often suggested. We consider a 3-component series system with exponentially-distributed component-failure times. This paper presents three types of approximate CIs based on: 1) asymptotic-normal theory for maximum likelihood estimators, 2) cube-root transformation of the exponential-distribution rate parameter, and 3) inverted likelihood-ratio tests. A simulation study assessed the coverage properties of these approximate CIs computed from censored & masked system failure data. The simulation results show that the asymptotic-normal theory CIs are too short and their lower and upper tail probabilities are asymmetric where there are few unmasked failures. The cube-root transformation of the exponential distribution rate parameter and inverted likelihood-ratio tests yield intervals with observed error probabilities close to the nominal ones. Given the relative ease of calculation, I recommend CIs based on the cube-root transformation. Various investigations have shown that likelihood-ratio CIs are much more appropriate for small samples than are the usual asymptotic-normal theory CIs. The present study shows that this finding holds when censored & masked failure data are used from systems with exponentially-distributed failure times. For this particular model, however, the obvious choice in practice is the CIs based on the transformation of the rate parameter. The performance of these CIs is comparable to that of the likelihood-ratio CIs, and the intervals themselves are much easier to calculate. Inverting likelihood-ratio tests, however, provides a more general method for CI estimation. I anticipate that it will work well for similar models involving lognormal or Weibull distributions for which parameter normalizing transformations are not as tractable.