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 x f of predictors when the regression model has the form Yi = m(xi)+ei where m is a function of xi 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 x f from a multivariate distribution. The nonparametric prediction region developed in this paper has correct asymptotic coverage if the data x1,..., xn are iid from a distribution with a nonsingular covariance matrix. For many distributions, this prediction region appears to have good coverage for n> 20p, 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