  This paper describes three methods for carrying out non-asymptotic inference
on partially identified parameters that are solutions to a class of
optimization problems. Applications in which the optimization problems arise
include estimation under shape restrictions, estimation of models of discrete
games, and estimation based on grouped data. The partially identified
parameters are characterized by restrictions that involve the unknown
population means of observed random variables in addition to structural
parameters. Inference consists of finding confidence intervals for functions of
the structural parameters. Our theory provides finite-sample lower bounds on
the coverage probabilities of the confidence intervals under three sets of
assumptions of increasing strength. With the moderate sample sizes found in
most economics applications, the bounds become tighter as the assumptions
strengthen. We discuss estimation of population parameters that the bounds
depend on and contrast our methods with alternative methods for obtaining
confidence intervals for partially identified parameters. The results of Monte
Carlo experiments and empirical examples illustrate the usefulness of our
method.
