Confidence Intervals and Tests of Hypotheses for the Reliability of a 2-Parameter Weibull System

A Monte Carlo simulation (2000 repetitions) is conducted to find empirically the standard deviation of the reliability of a system whose underlying distribution is a 2-parameter Weibull, for various values of the true reliability and the sample size n. The empirical values of the standard deviation are compared with values computed from an asymptotic formula based on the Cramér-Rao lower bound for the variance of an unbiased estimator. A general test of hypotheses developed in this paper is used for two hypothesized values of reliability, and uses the asymptotic standard deviations corresponding to the two values of reliability. Lower s-confidence limits for the true reliability are also calculated, again using the asymptotic standard deviation. These lower s-confidence limits agree closely with the results of a Monte Carlo simulation.