A note on tolerance regions for random vectors and best linear predictors

When a (p+q)-variate column vector (x′,y′)′ has a (p+q)-variate normal density with mean vector (μ₁,μ₂) and covariance matrix Ω, unknown, Schervish (1980) obtains prediction intervals for the linear functions of a future y, given x. He bases the prediction interval on the F-distribution. However, for a specified linear function the statistic to be used is Student's t, since the prediction intervals based on t are shorter than those based on F. Similar results hold for the multivariate linear regression model.