Constructing Optimal Prediction Intervals for Future Order Statistics

Prediction, by interval or point, of an unobserved random variable is a fundamental problem in statistics. This paper deals with constructing a prediction interval on a future observation Xr in an ordered sample of size n from an underlying distribution (under parametric uncertainty), where the first k observations X1 < X2 <  < Xk, 1k<rn, 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 optimisation of prediction intervals based on censored data emphasizes pivotal quantities relevant for obtaining ancillary statistics. It allows one to solve the optimisation problems in a simple way. An illustrative example is given.