Intelligent Constructing Exact Tolerance Limits for Prediction of Future Outcomes Under Parametric Uncertainty

Introduction

The logical purpose for a statistical tolerance limit (where 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 percentageγ) is to predict future outcomes for some production process which is treated as process, say, with stochastic variation of a product lifetime. The applications of tolerance limits (intervals) are varied. They included clinical and industrial applications, including quality control, applications to environmental monitoring, to the assessment of agreement between two methods or devices, and applications in industrial hygiene. For example, such tolerance limits are required, when planning life tests, engineers may need to predict the number of failures that will occur by the end of the test or to predict the amount of time that it will take for a specified number of units to fail. Tolerance limits of the type mentioned above are considered in this article, 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 extreme-value and two-parameter Weibull distributions under parametric uncertainty (when both parameters are unknown). 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 does not in need to make any assumption concerning the statistical functional form for the tolerance limit, is conceptually simple and easy to use. The scientific literature does not contain an analytical methodology for constructing exact γ-content tolerance limits with expected (1−α)-confidence on future order statistics coming from an extreme-value or Weibull distribution. One reason is that the theoretical concept and computational complexity of the tolerance limits is significantly more difficult than that of the standard confidence and prediction limits. However, in the literature there are several known methods for constructing (1-α)-prediction limits (in terms of this article, tolerance limits with expected (1-α)-confidence) on future order statistics coming from the two-parameter Weibull distribution. Therefore, finally, we give numerical examples, where the (1-α)-prediction limits obtained by using the known methods are compared with the results obtained through the proposed analytical methodology, which is illustrated in terms of the extreme-value and two-parameter Weibull distributions. Analytical formulas for the tolerance limits are available in the scientific literature for only simple cases, for example, for the upper or lower tolerance limit for a univariate normal population. Thus it becomes necessary to use new methods in order to derive exact statistical tolerance limits for many populations. The proposed, in this article, technique of intelligent constructing exact statistical γ-content tolerance limits with expected (1−α)-confidence, which are obtained here in terms of the two-parameter Weibull and extreme-value distributions, represents a novelty in the theory of statistical decisions.