Among all tests that are both unbiased and invariant under suitable groups under the assumptions of Problem 7.35, there exist UMP tests of

(i) H1 : α1 =···= αa = 0;

(ii) H2 : σ2 B/(nσ2 C + σ2) ≤ C;

(iii) H3 : σ2 C/σ2 ≤ C.

Note. The independence assumptions of Problems 7.35 and 7.36 often are not realistic. For alternative models, derived from more basic assumptions, see Scheffé (1956, 1959). Relations between the two types of models are discussed in Hocking (1973), Cohen and Miller (1976), and Stuart and Ord (1991).