## All Graduate Theses and Dissertations

5-1987

Thesis

#### Degree Name:

Master of Science (MS)

#### Department:

Mathematics and Statistics

#### Department name when degree awarded

Applied Statistics

Ronald Canfield

#### Abstract

A methodology which calculates a point estimate and confidence intervals for system reliability directly from component failure data is proposed and evaluated. This is a nonparametric approach which does not require the component time to failures to follow a known reliability distribution.

The proposed methods have similar accuracy to the traditional parametric approaches, can be used when the distribution of component reliability is unknown or there is a limited amount of sample component data, are simpler to compute, and use less computer resources. Depuy et al. (1982) studied several parametric approaches to calculating confidence intervals on system reliability. The test systems employed by them are utilized for comparison with published results. Four systems with sample sizes per component of 10, 50, and 100 were studied.

The test systems were complex systems made up of I components, each component has n observed (or estimated) times to failure. An efficient method for calculating a point estimate of system reliability is developed based on counting minimum cut sets that cause system failures.

Five nonparametric approaches to calculate the confidence intervals on system reliability from one test sample of components were proposed and evaluated. Four of these were based on the binomial theory and the Kolomogorov empirical cumulative distribution theory. 600 Monte Carlo simulations generated 600 new sets of component failure data from the population with corresponding point estimates of system reliability and confidence intervals. Accuracy of these confidence intervals was determined by determining the fraction that included the true system reliability.

The bootstrap method was also studied to calculate confidence interval from one sample. The bootstrap method is computer intensive and involves generating many sets of component samples using only the failure data from the initial sample. The empirical cumulative distribution function of 600 bootstrapped point estimates were examined to calculate the confidence intervals for 68, 80, 90 95 and 99 percent confidence levels.

The accuracy of the bootstrap confidence intervals was determined by comparison with the distribution of 600 point estimates of system reliability generated from the Monte Carlo simulations.

The confidence intervals calculated from the Kolomogorov empirical distribution function and the bootstrap method were very accurate. Sample sizes of 10 were not always sufficient for systems with reliabilities close to one.

#### Checksum

9629c0feb4d77ca65650a80f9cf78f1c

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