Construction of Prediction Intervals in Location‐Scale Families

Let x 1 ≤ x 2 ≤ x 3 … ≤ x r be the r smallest observations out of n observations from a location‐scale family with density \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{1}{\sigma}f\left({\frac{{x - \mu}}{\sigma}} \right) $\end{document} where μ and σ are the location and the scale parameters respectively. The goal is to construct a prediction interval of the form \documentclass{article}\pagestyle{empty}\begin{document}$ \left({\hat \mu + k_1 \hat \sigma,\,\hat \mu + k_2 \hat \sigma} \right) $\end{document} for a location‐scale invariant function, T(Y) = T(Y 1 , …, Y m ), of m future observations from the same distribution. Given any invariant estimators \documentclass{article}\pagestyle{empty}\begin{document}$ \hat \mu $\end{document} and \documentclass{article}\pagestyle{empty}\begin{document}$ \hat \sigma $\end{document} , we have developed a general procedure for how to compute the values of k 1 and k 2 . The two attractive features of the procedure are that it does not require any distributional knowledge of the joint distribution of the estimators beyond their first two raw moments and \documentclass{article}\pagestyle{empty}\begin{document}$ \hat \mu $\end{document} and \documentclass{article}\pagestyle{empty}\begin{document}$ \hat \sigma $\end{document} can be any invariant estimators of μ and σ. Examples with real data have been given and extensive simulation study showing the performance of the procedure is also offered.