Simultaneous Confidence Tubes in Multivariate Linear Regression

Simultaneous confidence bands have been shown in the statistical literature as powerful inferential tools in univariate linear regression. While the methodology of simultaneous confidence bands for univariate linear regression has been extensively researched and well developed, no published work seems available for multivariate linear regression. This paper fills this gap by studying one particular simultaneous confidence band for multivariate linear regression. Due to the shape of the band, the word ‘tube’ is more pertinent and so will be used to replace the word ‘band’. It is shown that the construction of the tube is related to the distribution of a largest eigenvalue. A simulation-based method is proposed to compute the 1−α quantile of this eigenvalue. With the computation power of modern computers the simultaneous confidence tube can be computed fast and accurately. A real data example is used to illustrate the method and many potential research problems have been pointed out.