61. Permitting interactions in the model of Problem 57 leads to the model X;jk = J.L + Ai + Bj + Cij + U;jk (i = 1, ... , a; j = 1, ... , b; k = 1, . . . , n) .

where the A's, B's, C's, and U's are independent normal with mean zero and variances a}, a~, ai, and a2• (i) Give an example of a situation in which such a model might be appropriate. (ii) Reduce the model to a convenient canonical form along the lines of Sections 5 and 8. (iii) Determine UMP unbiased tests of

(a) H\ : a~ = 0;

(b) H2 : ai = O