A New Approach to Constructing Tolerance Limits on Order Statistics in Future Samples Coming from a Normal Distribution

Abstract

Although the concept of statistical tolerance limits has been well recognized for long time, surprisingly, it seems that their applications remain still limited. Analytic formulas for the tolerance limits are available in only simple cases, for example, for the upper or lower tolerance limit for a univariate normal population. Thus it becomes necessary to use new or innovative approaches which will allow one to construct tolerance limits on future order statistics for many populations. In this paper, a new approach to constructing lower and upper tolerance limits on order statistics in future samples is proposed. Attention is restricted to invariant families of distributions under parametric uncertainty. The approach 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 past data are complete or Type II censored. The proposed approach requires a quantile of the F distribution and is conceptually simple and easy to use. For illustration, the normal distribution is considered. The discussion is restricted to one-sided tolerance limits. A practical example of finding a warranty assessment of image quality is given.