Maximum Likelihood Estimation, Exact Confidence Intervals for Reliability, and Tolerance Limits in the Weibull Distribution

This paper presents the results of a study of the maximum likelihood estimator, (t), of the reliability, R(t), when the two-parameter Weibull distribution is assumed. It is shown that the distribution of (t) depends only upon R(t) and n. It is observed that (t) is very nearly unbiased and has a variance that is practically equal to the Cramer-Rao lower bound for the variance of an unbiased estimator. Tables of lower confidence limits for the reliability are also provided. For an observed value of (t), the lower confidence limit can be read directly from the table for confidence levels of .75, 30, 35, .90, .95, and .98. The large sample normal approxmation for (t) is also investigated. Tolerance limits based on the maximum likelihood estimators of the Weibull parameters are developed. It is found that the tables needed for obtaining confidence interval for R(t) also enable one to obtain lower tolerance intervals. An example is given to help clarify the procedure.