Our discussion will concern primarily $\lambda$, the ratio of two variances which arise in discussing the "mixed" incomplete block model. In Section 1 we find first the class of invariant statistics for a test involving this ratio and second the joint distribution of these statistics. In Section 2 we use these statistics to construct a test (with certain optimum properties) of the hypothesis $\lambda < \lambda_0$ versus $\lambda > \lambda_1$.
"The Ratio of Variances in a Variance Components Model." Ann. Math. Statist. 26 (2) 325 - 329, June, 1955. https://doi.org/10.1214/aoms/1177728550