Tests for the homogeneity of a set of variances against ordered alternatives

Abstract

SuIMMAPRY Several statistics are investigated for testing the homogeneity of a set of variances of normal populations against ordered alternatives. The null distributions of these statistics are studied. Comparisons of powers of the tests are also made by Monte Carlo experiments. The test based on the modified Bartlett's statistic M* is found to perform best. For testing the homogeneity of variances of several normal distributions test statistics such as M by Bartlett (1937), Fmax by Hartley (1950) and G by Cochran (1941) have been proposed. These test statistics are appropriate when the alternative hypothesis is merely that the variances are not all equal. In this paper we consider the problem of testing against ordered alternatives. In ? 2 the modified M, the modified Fmax and the modified G statistics based on estimates of variances under order restrictions are introduced. We derive their null distributions. The approximate critical points of the modified M are computed for 3 to 12 populations by using the method due to Hartley (1939) for approximating the null distribution of M statistic. The critical points of the modified Fmax are computed for 3 to 6 populations. In ? 3 some comparisons of the powers of tests are made by computer simulation, including the conventional test statistics mentioned above, one proposed by Vincent (1961) and a modification of it.