Equality of More Than Two Variances and of More Than Two Dispersion Matrices Against Certain Alternatives

Abstract

In this paper, using the heuristic union-intersection principle [4], two tests are proposed, and the associated simultaneous confidence bounds on parametric functions which are measures of a certain type of departure from the respective null hypotheses are obtained. The first test is for the equality of $(k + 1)$ variances $(k \geqq 2)$ of $(k + 1)$ univariate normal populations, wherein we choose one of the variances as a standard (of course, unknown), and compare the other $k$ variances with it. The alternative to the hypothesis is that not all the $k$ variances are equal to the standard one. The proposed test may be called the simultaneous variance ratios test. The well-known Hartley's $F_{\max}$ test [2] for the case of equal sample sizes is not equivalent to the present test even when all samples are of the same size since the alternatives in the two cases are different. In the alternative in Hartley's test, aside from the inequality of the $k$ variances to the standard one, the mutual inequality of the $k$ variances also plays an important role. The second test proposed in this paper, is a multivariate extension of the first. This paper also considers the distribution problems that arise in connection with both the tests. The nonavailability of tables at the moment makes the immediate practical application of the tests and the associated confidence bounds not possible. Sections 1, 2, and 3 deal with the univariate problem and Sections 4 and 5 deal with the multivariate extensions.