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

Abstract

Consider a general linear model with p -dimensional parameter vector beta and i.i.d. normal errors. Let K(1), ..., K(k ), and L be linearly independent vectors of constants such that L(T)beta not equal 0. We describe exact simultaneous tests for hypotheses that Ki(T)beta/L(T)beta 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.