Lower bound on reliability for Weibull distribution when shape parameter is not estimated accurately

The mathematical relationships between the shape parameter β and estimates of reliability and a life limit lower bound for the two-parameter Weibull distribution are discussed. It is shown that under general conditions both the reliability lower bound and the allowable life limit lower bound (tolerance limit) have unique global minima over a range of β. Hence, lower bound solutions can be obtained without assuming or estimating β. The existence and uniqueness of these lower bounds are proven. Some data examples are given to show how these lower bounds can be easily established and to demonstrate their practicality. The method has proven to be useful when using the Weibull distribution in analysis of no-failure or few-failure data. The results are widely applicable, not only in the aerospace industry, but anywhere that system reliabilities are high