Pertinent Prediction Intervals in Linear Regression

In linear regression, a point predictor Y ^ f of a future response Y f associated with a regressor value of interest x ̲ f can easily be constructed. Since Y ^ f will always incur a prediction error, it is desirable to accompany the point predictor by a prediction interval, say C ( x ̲ f ) , that will contain the target Y f with a pre-specified high probability, e.g., 90%. An estimated prediction interval, say C ^ ( x ̲ f ) , is called pertinent if its construction incorporates the variability of all estimators that are employed in the prediction problem. So far, pertinent prediction intervals have only been constructed via some form of bootstrap. However, resampling can be quite computationally expensive since the estimation/prediction problem has to be re-calculated on a large number of pseudo-scatterplots, each having the same sample size as the original one. The paper at hand proposes a short-cut that directly employs the asymptotic normal distribution of relevant estimators—as opposed to a bootstrap histogram—in order to capture their variability. The resulting prediction interval achieves pertinence without full-scale resampling, thus offering computational savings of orders of magnitude.