  This paper derives the rate of convergence and asymptotic distribution for a
class of Kolmogorov-Smirnov style test statistics for conditional moment
inequality models for parameters on the boundary of the identified set under
general conditions. In contrast to other moment inequality settings, the rate
of convergence is faster than root-$n$, and the asymptotic distribution depends
entirely on nonbinding moments. The results require the development of new
techniques that draw a connection between moment selection, irregular
identification, bandwidth selection and nonstandard M-estimation. Using these
results, I propose tests that are more powerful than existing approaches for
choosing critical values for this test statistic. I quantify the power
improvement by showing that the new tests can detect alternatives that converge
to points on the identified set at a faster rate than those detected by
existing approaches. A monte carlo study confirms that the tests and the
asymptotic approximations they use perform well in finite samples. In an
application to a regression of prescription drug expenditures on income with
interval data from the Health and Retirement Study, confidence regions based on
the new tests are substantially tighter than those based on existing methods.
