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This paper shows that moment inequality tests that are asymptotically similar on the boundary of the null hypothesis exist, but have very poor power. Hence, existing tests in the literature, which are asymptotically non-similar on the boundary, are not deﬁcient. The results are obtained by ﬁrst establishing results for the ﬁnite-sample multivariate normal one-sided testing problem. Then, these results are shown to have implications for more general moment inequality tests that are used in the literature on partial identiﬁcation.
Andrews, Donald W.K., "Similar-on-the-Boundary Tests for Moment Inequalities Exist, But Have Poor Power" (2011). Cowles Foundation Discussion Papers. 2163.