Adequate Mathematical Models of the Cumulative Distribution Function of Order Statistics to Construct Accurate Tolerance Limits and Confidence Intervals of the Shortest Length or Equal Tails

The technique used here emphasizes pivotal quantities and ancillary statistics relevant for obtaining tolerance limits (or confidence intervals) for anticipated outcomes of applied stochastic models under parametric uncertainty 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 practical numerical examples, where the proposed analytical methodology is illustrated in terms of the exponential distribution. Applications to other log-location-scale distributions could follow directly.