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.
Armstrong, Timothy B., "A Note on Minimax Testing and Confidence Intervals in Moment Inequality Models" (2014). Cowles Foundation Discussion Papers. 2390.