Optimization of prediction intervals for order statistics based on censored data

Prediction intervals for order statistics are widely used for reliability problems and other related problems. The determination of these intervals has been extensively investigated. But the optimality property of these intervals has not been fully explored. In this paper, in order to discuss this problem, a risk function is introduced to compare prediction intervals. In particular, new-sample prediction based on a previous sample (i.e., when for predicting the future observation in a new sample there are available the data only from a previous sample), and within-sample prediction based on the early observed data from a current experiment (i.e., when for predicting the future observation in a sample there are available the early observed data only from that sample). We restrict attention to families of distributions invariant under location and/or scale changes. The technique used here for optimization of prediction intervals based on censored data emphasizes pivotal quantities relevant for obtaining ancillary statistics. It allows one to solve the optimization problems in a simple way. An illustrative example is given.