Counterexample. The following example shows that the equivariance of S(x) assumed in the paragraph following Lemma 6.11.1 does not follow from the other assumptions of this lemma. In Example 6.5.1, let n = 1, let G(1)

be the group G of Example 6.5.1, and let G(2) be the corresponding group when the roles of Z and Y = Y1 are reversed. For testing H(θ0) : θ = θ0 against θ = θ0 let Gθ0 be equal to G(1) augmented by the transformation Y
= θ0 − (Y1 − θ0)

when θ ≤ 0, and let Gθ0 be equal to G(2) augmented by the transformation Z
= θ0 − (Z − θ0) when θ > 0. Then there exists a UMP invariant test of H(θ0)

under Gθ0 for each θ0, but the associated confidence sets S(x) are not equivariant under G = {Gθ, −∞ <θ< ∞}.