Prediction Intervals for Future Observations Based on Samples of Random Sizes

On the basis of failure times of an informative sample (X-sample) of random size M of iid continuous random variables, we consider the prediction problem of failure times of another independent future sample (Y-sample) of random size N of iid variables from the same distribution. In this paper, we derive various exact prediction intervals for future failure times from the Y -sample based on the failure times from the X-sample. Specifically, prediction intervals for individual failure times as well as outer and inner prediction intervals are derived based on X-order statistics. Prediction intervals for the increments of order statistics are also investigated. Exact expressions for the coverage probabilities of these intervals are also derived and computed numerically. A practical example on a biometric data set is used to illustrate the results developed here.