Asymptotically Optimal Regression Prediction Intervals and Prediction Regions for Multivariate Data

摘要

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.