Confidence Bounds on Vector Analogues of the "Ratio of Means" and the "Ratio of Variances" for Two Correlated Normal Variates and Some Associated Tests

Abstract

In this paper confidence bounds are obtained (i) on the ratio of variances of a (possibly) correlated bivariate normal population, and then, by generalization, (ii) on a set of parametric functions of a (possibly) correlated $p + p$ variate normal population, which plays the same role for a $2p$-variate population as the ratio of variances does for the bivariate case, (iii) on the ratio of means of the population indicated in (i), and, by generalization, (iv) on a set of parametric functions of the population indicated in (ii), which plays the same role for this problem as the ratio of means does for the bivariate case. For (i) and (iii) the confidence coefficient is any preassigned $1 - \alpha$ and the distribution involved is the central $t$-distribution, while for (ii) and (iv), the confidence statement in each case is a simultaneous one with a joint confidence coefficient greater than or equal to a preassigned $1 - \alpha$. For (ii) the distribution involved is that of the central largest canonical correlation coefficient (squared), and for (iv) the distribution involved is that of the central Hotelling's $T^2$. As far as the authors are aware the results on (ii) and (iv) are new and so perhaps that on (i). But the result on (iii) has been in the field for a long time in various superficially different forms. An important point to keep in mind on these problems is that, for such confidence bounds and the associated tests of hypotheses to be physically meaningful, the two variates for the bivariate distribution should be comparable. For example, they might refer to the same characteristic of a set of individuals before and after a feed. Likewise, for a $(p + p)$-variate distribution, the $p$ variates of the first set should be comparable to $p$ variates of the second set. For example, they might refer to several characteristics of a set of individuals before and after a treatment. In each case the confidence bounds are obtained by inverting the test of a certain hypothesis, which is indicated at its proper place. Thus, for the $(p + p)$-variate problem, we assume that there are $p$ pairs of comparable variates and it is the pairwise comparison for these $p$ pairs that seems, in this situation, to be physically more meaningful than anything else. Any general bounds that will be obtained in this paper are to be regarded, in a large measure, as a means to this end, although there could conceivably be physical questions, some of which will be illustrated in a later applied paper to be published elsewhere, to which these more general bounds would be pertinent.