A confidence band simultaneously provides useful information within the acceptable range of the unknown regression model, and different confidence bands can often be constructed for the same regression model. For a simple regression line, Lieuheiter (2007) proposed using the confidence set level corresponding to a confidence band as an optimality criterion in comparing confidence bands; the smaller the confidence set level, the better the corresponding confidence band. In the study of confidence bands for a multiple linear regression model, this minimum-level confidence set criterion can be extended to a minimum-volume confidence set criterion.
In this paper, hyperbolic and fixed-width confidence bands for a multiple linear regression model are compared over a specific ellipse of the predictor variables under test. It can be seen that the better performance of one band than another depends on the value of a certain angle that determines the size of the predictor variable region. When this angle, and therefore the size of the predictor variable region, is small, the fixed-width confidence band is better than the hyperbolic confidence band, but only marginally. When this angle, and therefore the size of the predictor variable region, is large, the hyperbolic confidence band can be better than the fixed-width confidence band overall.