Simultaneous Inference for Ratios of Linear Combinations of General Linear Model Parameters

Consider a general linear model with p ‐dimensional parameter vector β and i.i.d. normal errors. Let K 1 , …, K k , and L be linearly independent vectors of constants such that L T β ≠ 0. We describe exact simultaneous tests for hypotheses that K $ _i ^T $ β / L T β equal specified constants using one‐sided and two‐sided alternatives, and describe exact simultaneous confidence intervals for these ratios. In the case where the confidence set is a single bounded contiguous set, we describe what we claim are the best possible conservative simultaneous confidence intervals for these ratios – best in that they form the minimum k ‐dimensional hypercube enclosing the exact simultaneous confidence set. We show that in the case of k = 2, this “box” is defined by the minimum and maximum values for the two ratios in the simultaneous confidence set and that these values are obtained via one of two sources: either from the solutions to each of four systems of equations or at points along the boundary of the simultaneous confidence set where the correlation between two t variables is zero. We then verify that these intervals are narrower than those previously presented in the literature. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)