Multiple-use prediction and calibration for all future values: Exact simultaneous tolerance bands for multiple and polynomial regressions.

Multiple-use prediction and calibration for all future values play an important role in many areas, including health and medical research. Simultaneous tolerance bands (STBs) can be employed for these purposes. This article studies the construction of exact STBs for multiple regression over any given rectangular covariate regions and for polynomial regression over any given covariate intervals. We first fill the gap in constructing the exact STBs for multiple regression over rectangular regions. We then propose a general form of STBs and develop a method for computing the critical constants for all STBs considered, including existing ones in the literature. A new STB is also proposed for both multiple and polynomial regressions. This new STB and several existing forms are then compared under the average shift (AS) criterion. Numerical results indicate that the new STB performs best under the AS criterion and is therefore recommended. In addition, a computationally efficient algorithm is presented. Real-life examples are provided for illustration.