On Confidence Limits for the Reliability of Systems

The asymptotic chi-square distribution of the log-likelihood ratio is used to obtain approximate confidence intervals for the reliability of any system which may be represented by a monotone function of Bernoulli variates. This generalizes the results of A. Madansky, Technometrics, November 1965, for series, parallel and series-parallel systems. The method used is to parameterize the log-likelihood equation so as to find the interval of parameter values which keeps the log-likelihood less than or equal to the specified quantile of the chi-square distribution. This is done by introducing an operator depending upon the parameter, a fixed point of which is the solution of the likelihood ratio equation, and by showing the operator is a contractive map and hence has a unique fixed point depending continuously on the parameter. The solution can be found simply by iteration.