A New Pivot-Based Approach to Constructing Prediction Limits and Shortest-Length or Equal Tails Confidence Intervals for Future Outcomes under Parametric Uncertainty

It is often desirable to have statistical prediction limits available for future outcomes from the distributions used to describe time-to- failure data in reliability problems. For example, one might wish to know if at least a certain proportion, say y, of a manufactured product will operate at least t hours. This question cannot usually be answered exactly, but it may be possible to determine a lower prediction limit L(X), based on a random sample X, such that one can say with a certain confidence - (1- a) that at least 100% of the product will operate longer than L(X). Then reliability statements can be made based on L(X), or, decisions can be reached by comparing L(X) to t. Predictions limits of the type mentioned above are considered in this paper. A new approach is used to construct unbiased prediction limits and shortest-length or equal tails confidence intervals for future outcomes under parametric uncertainty of the underlying distributions through pivot-based estimates of these distributions. The approach isolates and eliminates unknown parameters of the reliability problem and uses the past statistical data as efficiently as possible. Unlike the Bayesian approach, the proposed approach is independent of the choice of priors and represents a novelty in the theory of statistical decisions. It allows one to eliminate unknown parameters from the problem and to find the efficient statistical decision rules, which often have smaller risk than any of the well-known decision rules. To illustrate the proposed approach, some practical applications are given.