24. Let XI' X; be independent normal variables with variance 1 and means ~I' ...
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24. Let XI"'" X; be independent normal variables with variance 1 and means ~I' • • • , ~n' and consider the problem of testing H : ~I = . .. = ~n = 0 against the alternatives K = {KI , • •• , Kn } , where K;: ~j = 0 for j '" i, ~; = (known and positive). Show that the problem remains invariant under permutation of the X's and that there exists a UMP invariant test 4>0 which rejects when Ee-(Xj > C, by the following two methods. 534 (i) The order statistics X(1) < . . . < X(n) constitute a maximal invariant. (ii) Let 10 and h denote the densities under H and K; respectively. Then the level-a test 4>0 of H vs. K' :1 = (ljn)Eh is UMP invariant for testing H vs. K. [(ii): If 4>0 is not UMP invariant for H vs. K, there exists an invariant test 4>1 whose (constant) power against K exceeds that of 4>0' Then 4>1 is also more powerful against K'.]
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