Optimal prediction intervals for order statistics coming from location-scale families

Prediction, by interval or point, of an unobserved random variable is a fundamental problem in statistics. This paper deals with obtaining a prediction interval on a future observation Xl in an ordered sample of size m from an underlying distribution, which belongs to the location-scale family of distributions, for the situation where the first k observations X₁ < X₂ <... < X_k, 1≤k<l≤m, have been observed. Prediction intervals for future order statistics are widely used for reliability problems and other related problems. But the optimality property of these intervals has not been fully explored. To compare prediction intervals, we introduce a piecewise-linear loss function. The interval which minimizes a risk, associated with this piecewise-linear loss function, among the class of invariant prediction intervals is obtained. 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. Illustrative examples are given for the Gumbel and two-parameter exponential distributions. The results can be also applied to related distributions.