52. Counterexample. The following example shows that the equivariance of S(x) assumed in the paragraph following Lemma 5 does not follow from the other assumptions of this lemma. In Example 8, let n = 1, let G(1) be the group G of Example 8, and let G(2) be the corresponding group when the roles of Z and Y = Yl are reversed. For testing H(80 ) : 8 = 80 against 8 '" 80 let G90 be equal to GO) augmented by the transformation Y' = 80 - (YI - 80 ) when 8 0, and let G90 be equal to G(2) augmented by the transformation Z' = 80 - (Z - 80 ) when 8> 0. Then there exists a UMP invariant test of H(80 ) under G90 for each 80 , but the associated confidence sets S(x) are not equivariant under G = {Gs, - 00 < 8 < oo}.