Asymptotically Optimal Regression Prediction Intervals and Prediction Regions for Multivariate Data

This paper presents asymptotically optimal prediction intervals and prediction regions. The prediction intervals are for a future response Yf given a p × 1 vector f of predictors when the regression model has the form Yᵢ = m(ᵢ) + eᵢ where m is a function of ᵢ and the errors eᵢ are iid from a continuous unimodal distribution. The prediction intervals have coverage near or higher than the nominal coverage for many techniques even for moderate sample size n, say $n >$ 10(model degrees of freedom). The prediction regions are for a future vector of measurements f from a multivariate distribution. The nonparametric prediction region developed in this paper has correct asymptotic coverage if the data ₁, ..., ₙ are iid from a distribution with a nonsingular covariance matrix. For many distributions, this prediction region appears to have good coverage for $n > 20 p$, and this region is asymptotically optimal on a large class of elliptically contoured distributions. Hence the prediction intervals and regions perform well for moderate sample sizes as well as asymptotically.