On Confidence Limits for the Reliability of Systems
0. Summary. 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 chisquare 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. 1. Introduction. Several approaches have been made to the problem of determining confidence limits for system reliability. Buehler [2] has given upper confidence lirnits for the product of two binomial parameters using Poisson approximations with an accuracy believed to be adequate whenever both sample sizes exceed forty. Buehler's method has been extended by Lipow [5] and Steck [8], for certain cases and short tabulations have been made. Madansky [6] has made use of the log-likelihood ratio's asymptotic chi-square distribution to obtain confidence intervals for series, parallel and series-parallel systems. It is the extension of this last result to a much wider class of systems that we present here. The reason for this effort is the acknowledged inadequacy in many practical instances of the only known method of finding confidence bounds, from data on the performance of the components, for the reliability of general systems. This known method is the use of the asymptotic normality of the maximum likelihood estimates. A comparison, for certain cases of practical importance, of the method presented here with this alternative method is carried out in [7]. There the authors have shown that the use of the asymptotic chi-square method to determine confidence limits yields results which are substantially better for small sample sizes and equivalent for large ones for the cases considered. In order to be specific about the class of systems which we consider here, it is necessary to introduce some notation. For the ith component among m, let the Bernoulli random variable Yi indicate performance by taking the value one for success and zero for failure. The state of the components then is the vector Y = ( Y1, * * *, Yin). By a coherent (monotone) structure we follow [1] to mean there