63. Among all tests that are both unbiased and invariant under suitable groups under the assumptions of Problem 62, there exist UMP tests of (i) HI : a\ = . . . = a" = 0; (ii) H2 + ( 2 ) s C; (iii) H3 : a2/a2 s C. Note. The independence assumptions of Problems 62 and 63 often are not realistic. For alternative models, derived from more basic assumptions, see Scheffe (1956, 1959). Relations between the two types of models are discussed in Hocking (1973), Cohen and Miller (1976), and Kendall, Stuart, and Ord (1983).