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This paper derives asymptotic approximations to the power of Cramer-von Mises (CvM) style tests for inference on a ﬁnite dimensional parameter deﬁned by conditional moment inequalities in the case where the parameter is set identiﬁed. Combined with power results for Kolmogorov-Smirnov (KS) tests, these results can be used to choose the optimal test statistic, weighting function and, for tests based on kernel estimates, kernel bandwidth. The results show that, in the setting considered here, KS tests are preferred to CvM tests, and that a truncated variance weighting is preferred to bounded weightings.
Armstrong, Timothy B., "On the Choice of Test Statistic for Conditional Moment Inequalities" (2014). Cowles Foundation Discussion Papers. 2371.