Asymptotically Valid Prediction Intervals for Linear Models

Abstract

For estimating the parameters of a linear model, whether by least squares or many robust alternatives, one can use confidence limits that are based on large-sample normal approximations. These limits are asymptotically valid for a wide range of error distributions. Unfortunately, asymptotic validity does not automatically hold for prediction limits for a new observation, because they depend on both parameter estimates and the underlying error distribution. Resampling methods provide prediction limits that are asymptotically valid, but they tend to be computationally intensive, especially when used with robust estimators. This article presents alternative prediction limits that are asymptotically valid, are not based on resampling, and are computationally more tractable than methods that are. They are based on quantiles of a convolution of the empirical distribution of the residuals and the limiting normal distribution of the parameter estimates. Simulations suggest that the proposed limits perform reasona...