Finding Unbiased Simultaneous Prediction Limits for Order Statistics of Future Samples with Applications

Abstract

This paper provides procedures for finding unbiased simultaneous prediction limits on the observations or functions of observations of all of k future samples using the results of a previous sample from the same underlying distribution belonging to invariant family. The results have direct application in reliability theory, where the time until the first failure in a group of several items in service provides a measure of assurance regarding the operation of the items. The simultaneous prediction limits are required as specifications on future life for components, as warranty limits for the future performance of a specified number of systems with standby units, and in various other applications. Prediction limit is an important statistical tool in the area of quality control. The lower simultaneous prediction limits are often used as warranty criteria by manufacturers. The initial sample and k future samples are available, and the manufacturer wants to have a high assurance that all of the k future orders will be accepted. It is assumed throughout that k + 1 samples are obtained by taking random samples from the same population. In other words, the manufacturing process remains constant. The results in this paper are generalizations of the usual prediction limits on observations or functions of observations of only one future sample. In the paper, attention is restricted to invariant families of distributions. The technique 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. Applications of the proposed procedures are given for the two-parameter exponential and Weibull distributions. The exact prediction limits are found and illustrated with a numerical example.