Construction of Prediction Intervals in Location-Scale Families

Let x₁ ≤x₂ ≤x₃ … ≤x_r be the r smallest observations out of n observations from a location-scale family with density $ \frac{1}{\sigma}f\left({\frac{{x - \mu}}{\sigma}} \right) $ where μ and σ are the location and the scale parameters respectively. The goal is to construct a prediction interval of the form $ \left({\hat \mu + k_1 \hat \sigma,\,\hat \mu + k_2 \hat \sigma} \right) $ for a location-scale invariant function, T(Y) = T(Y₁, …, Y_m), of m future observations from the same distribution. Given any invariant estimators $ \hat \mu $ and $ \hat \sigma $, we have developed a general procedure for how to compute the values of k₁ and k₂. 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 $ \hat \mu $ and $ \hat \sigma $ 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.