Constructing unbiased prediction limits on future outcomes under parametric uncertainty of underlying models via pivotal quantity averaging approach

Abstract

This paper presents a new simple, efficient and useful technique for constructing lower and upper unbiased prediction limits on outcomes in future samples under parametric uncertainty of underlying models. For instance, consider a situation where such limits are required. A customer has placed an order for a product which has an underlying time-to-failure distribution. The terms of his purchase call for k monthly shipments. From each shipment the customer will select a random sample of q units and accept the shipment only if the smallest time to failure for this sample exceeds a specified lower limit. The manufacturer wishes to use the results of an experimental sample of n units to calculate this limit so that the probability is γ that all k shipments will be accepted. It is assumed that the n experimental units and the kq future units are random samples from the same population. In this paper, attention is restricted to invariant families of distributions. The pivotal quantity averaging approach used here emphasizes pivotal quantities relevant for obtaining ancillary statistics 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 pivotal quantity averaging approach is conceptually simple and easy to use. For illustration, a left-truncated Weibull, two-parameter exponential, and Pareto distribution are considered. A practical numerical example is given.