Testing the equality of a large number of normal population means

It is challenging to consider the problem of testing the equality of normal population means when the number of populations is large compared to the sample sizes. In ANOVA with the assumption of homogeneous variance, the FF-test is known as an exact test. When variances are heterogeneous, due to the complication, there are various tests with only approximate forms–either approximate chi-square or approximate FF-test. Two types of tests are proposed with their asymptotic normality as the number of population increases. pp-values from those tests are adjusted based on higher order asymptotics such as Edgeworth expansion so that the proposed tests can be considered even for moderate values of kk. Numerical studies including simulations and real data examples are presented with comparison to existing tests.