Under the assumptions of Problem 6.70, suppose that a family of confidence sets S(x) is equivariant under G∗. Then there exists a set B in the range space of the pivotal V such that (6.72) holds. In this sense, all equivariant confidence sets can be obtained from pivotals.

[Let A be the subset of X × w given by A = {(x, θ) : θ ∈ S(x)}. Show that gA˜ = A, so that any orbit of G˜ is either in A or in the complement of A. Let the maximal invariant V (x, θ) be represented as in Section 6.2 by a uniquely defined point on each orbit, and let B be the set of these points whose orbits are in A.

Then V (x, θ) ∈ B if and only if (x, θ) ∈ A.] Note.