  This note uses a simple example to show how moment inequality models used in
the empirical economics literature lead to general minimax relative efficiency
comparisons. The main point is that such models involve inference on a low
dimensional parameter, which leads naturally to a definition of "distance"
that, in full generality, would be arbitrary in minimax testing problems. This
definition of distance is justified by the fact that it leads to a duality
between minimaxity of confidence intervals and tests, which does not hold for
other definitions of distance. Thus, the use of moment inequalities for
inference in a low dimensional parametric model places additional structure on
the testing problem, which leads to stronger conclusions regarding minimax
relative efficiency than would otherwise be possible.
