25. The UMP invariant test 4>0 of Problem 24 (i) maximizes the minimum power over K; (ii) is admissible. (iii) For testing the hypothesis H of Problem 24 against the alternatives K' = {KI , ... , Kn , K{,.. . , K~}, where under Ks : = 0 for all j'" i, = -~ determine the UMP test under a suitable group G', and show that it is both maximin and invariant. [ii): Suppose 4>' is uniformly at least as powerful as 4>0' and more powerful for at least one K;, and let 4>*( XI" '" x n ) = E4>'( x;" ... , x;) n! ' where the summation extends over all permutations. Then 4>* is invariant, and its power is independent of i and exceeds that of 4>0 ']