In the multivariate calibration problem using a multivariate linear model, an exact confidence region is constructed. It is shown that the region is always nonempty and is invariant under nonsingular transformations.
"An Exact Confidence Region in Multivariate Calibration." Ann. Statist. 22 (1) 94 - 105, March, 1994. https://doi.org/10.1214/aos/1176325359