Construction of a Conservative Confidence Region from Projections of an Exact Confidence Region in Multiple Linear Regression

The problem of constructing a confidence region for simultaneously estimating p, p ≥ 2, linear regression parameters for which confidence statements can be made on the individual parameters is revisited. Here, an intercept may be included among the p paremeters. The technique is due to Working and Hotelling (1929) and Scheffè (1959) and uses the p separate projections of the exact (1 – α) 100% confidence ellipsoid (ellipse if p = 2) to give confidence intervals for each regression parameter. The Cartesian product of these p confidence intervals gives a p-dimensional rectangle that contains the confidence ellipsoid and hence has a joint confidence coefficient of at least (1–α). A simple calculus proof is given to determine these projections. The projection procedure is compared with the Bonferroni procedure for this case.