A class of invariant hypothesis tests is considered for the purpose of testing a variance ratio arising in mixed models. Members of the class are most powerful for specific alternatives and limiting members of the class are Wald's test and the locally most powerful test. It is demonstrated that the locally most powerful test has the highest and Wald's test has the lowest asymptotic power when an asymptotically unbalanced sequence of ANOVA designs is considered under Pitman alternatives.
"Power Comparisons for Invariant Variance Ratio Tests in Mixed Anova Models." Ann. Statist. 17 (1) 318 - 326, March, 1989. https://doi.org/10.1214/aos/1176347019