Before revision:  The unit-wise sets give possible group memberships for a given unit and the joint sets give possible vectors of group memberships for all units . We also propose an algorithm that can improve the power of our procedures by detecting units that are easy to classify. The confidence sets invert a test for group membership that is based on a characterization of the true group memberships by a system of moment inequalities. To construct the joint confidence, we solve a high-dimensional testing problem that tests group membership simultaneously for all units .
After revision:  The unit-wise sets give possible group memberships for a given unit and the joint sets give possible vectors of group memberships for all units with pre-specified probability and is constructed by inverting many simultaneous unit-specific one-sided tests for group membership .