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 Scheff´e

(1956, 1959). Relations between the two types of models are discussed in Hocking

(1973), Cohen and Miller (1976), and Stuart and Ord (1991).