The Uniformly More Powerful Tests than the Likelihood Ratio Test Using Intersection-Union Hypotheses for Variance of Independent Samples from Normal Distribution

Results and DiscussionAlthough LRTs are the most common testing methods, they are not optimal for constrained hypotheses, including comparison between the variances in normal populations, discussed in this article.IUT tests have more power for unequal samples and the same power for equal samples.Therefore, comparing these two tests, IUT can be recommended than LRTs.Both LRTs and IUTs have less power than R and S tests.If it is possible to use both R and S tests, they are preferred over the previous tests.In comparing the two R and S tests, this study's results also show the slight superiority of the S test over the R test.The result in other distributions and other parameters depends on future research results.A significant weakness in the two R and S tests can be attributed to their distinct and non-monotonicity rejection regions.In a monotone test, the rejection regions are nested; therefore, not all points farther from the critical points can be considered to belong to the rejection region.In such circumstances, finding the rejection regions required complicated calculations.Until research has reached simple calculations in these two tests, it is more desirable to use IUT.Achieving UMP or at least unbiased tests for such hypotheses can be a turning point in statistical tests. ConclusionThis article presents an approach to constructing size-alpha tests that are more powerful than LRT.The union-intersection test for the sign test of variances in several normal distributions is proposed and compared with the LRT.Although the union-intersection test is more powerful, neither test is unbiased.Two rectangular and smoothed tests have been examined for a more powerful test.