## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 39, Number 5 (1968), 1463-1472.

### On Confidence Limits for the Reliability of Systems

Janet M. Myhre and Sam C. Saunders

#### Abstract

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.

#### Article information

**Source**

Ann. Math. Statist., Volume 39, Number 5 (1968), 1463-1472.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177698125

**Digital Object Identifier**

doi:10.1214/aoms/1177698125

**Mathematical Reviews number (MathSciNet)**

MR232517

**Zentralblatt MATH identifier**

0193.17801

**JSTOR**

links.jstor.org

#### Citation

Myhre, Janet M.; Saunders, Sam C. On Confidence Limits for the Reliability of Systems. Ann. Math. Statist. 39 (1968), no. 5, 1463--1472. doi:10.1214/aoms/1177698125. https://projecteuclid.org/euclid.aoms/1177698125