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This paper presents asymptotically optimal prediction intervals and prediction regions. The prediction intervals are for a future response $Y_f$ given a $p imes 1$ vector $x_f$ of predictors when the regression model has the form $Y_i = m(x_i) + e_i$ where $m$ is a function of $x_i$ and the errors $e_i$ 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 $x_f$ from a multivariate distribution. The nonparametric prediction region developed in this paper has correct asymptotic coverage if the data $x_1, ..., x_n$ 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

Asymptotically Optimal Regression Prediction Intervals and Prediction Regions for Multivariate Data