81. Under the assumptions of Problem 79, 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 (70) holds. In this sense, all equivariant confidence sets can be obtained from pivotals. [Let A be the subset of flEx w given by A = {(x, 8): 8 E S(x)}. Show that gA = A, so that any orbit of Gis either in A or in the complement of A. Let the maximal invariant V( x, 8) be represented as in Section 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 , 8) E B if and only if (x, 8) EA.] Note. Problem 80 provides a simple check of the equivariance of confidence sets. In Example 21, for instance, the confidence sets (41) are based on the pivotal vector (XI - ~I" .. , X, - t), and hence are equivariant.