A New Technique for Intelligent Constructing Exact γ-content Tolerance Limits with Expected (1 – α)-confidence on Future Outcomes in the Weibull Case Using Complete or Type II Censored Data

Abstract

The logical purpose for a statistical tolerance limit is to predict future outcomes for some (say, production) process. The coverage value γ is the percentage of the future process outcomes to be captured by the prediction, and the confidence level (1 – α) is the proportion of the time we hope to capture that percentage γ. Tolerance limits of the type mentioned above are considered in this paper, which presents a new technique for constructing exact statistical (lower and upper) tolerance limits on outcomes (for example, on order statistics) in future samples. Attention is restricted to the two-parameter Weibull distribution under parametric uncertainty. The technique used here emphasizes pivotal quantities relevant for obtaining tolerance factors and is applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. It does not require the construction of any tables and is applicable whether the experimental data are complete or Type II censored. The exact tolerance limits on order statistics associated with sampling from underlying distributions can be found easily and quickly making tables, simulation, Monte Carlo estimated percentiles, special computer programs, and approximation unnecessary. The proposed technique is based on a probability transformation and pivotal quantity averaging. It is conceptually simple and easy to use. The discussion is restricted to one-sided tolerance limits. Finally, we give numerical examples, where the proposed analytical methodology is illustrated in terms of the two-parameter Weibull distribution. Applications to other log-location-scale distributions could follow directly.