  This paper considers inference for a function of a parameter vector in a
partially identified model with many moment inequalities. This framework allows
the number of moment conditions to grow with the sample size, possibly at
exponential rates. Our main motivating application is subvector inference,
i.e., inference on a single component of the partially identified parameter
vector associated with a treatment effect or a policy variable of interest.
  Our inference method compares a MinMax test statistic (minimum over
parameters satisfying $H_0$ and maximum over moment inequalities) against
critical values that are based on bootstrap approximations or analytical
bounds. We show that this method controls asymptotic size uniformly over a
large class of data generating processes despite the partially identified many
moment inequality setting. The finite sample analysis allows us to obtain
explicit rates of convergence on the size control. Our results are based on
combining non-asymptotic approximations and new high-dimensional central limit
theorems for the MinMax of the components of random matrices. Unlike the
previous literature on functional inference in partially identified models, our
results do not rely on weak convergence results based on Donsker's class
assumptions and, in fact, our test statistic may not even converge in
distribution. Our bootstrap approximation requires the choice of a tuning
parameter sequence that can avoid the excessive concentration of our test
statistic. To this end, we propose an asymptotically valid data-driven method
to select this tuning parameter sequence. This method generalizes the selection
of tuning parameter sequences to problems outside the Donsker's class
assumptions and may also be of independent interest. Our procedures based on
self-normalized moderate deviation bounds are relatively more conservative but
easier to implement.
