Asymptotic normality and uncertainty bounds for reliability estimates from subsystem and full system tests

A previous paper (Spall, 2010) described a method for estimating the reliability of a complex system based on a combination of full system and subsystem tests. A maximum likelihood estimate (MLE) is formed to estimate the subsystem reliabilities and the full system reliability. While the previous paper gave conditions under which the MLE converges to the true reliability as the sample size gets large, the reference left open the question of how to formally compute uncertainty bounds (e.g., confidence bounds). A key part of computing such bounds is the determination of the large-sample (asymptotic) distribution for the estimate. In addition, the asymptotic distribution is important for determining whether the data have enough information to provide meaningful estimates of full system and subsystem reliabilities. This paper presents formal conditions for the asymptotic normality of the MLE to the true full system and subsystem reliability values. The paper also discusses a Monte Carlo-based bootstrap method for computing uncertainty bounds.