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This paper describes three methods for carrying out non-asymptotic inferenceon partially identified parameters that are solutions to a class ofoptimization problems. Applications in which the optimization problems ariseinclude estimation under shape restrictions, estimation of models of discretegames, and estimation based on grouped data. The partially identifiedparameters are characterized by restrictions that involve the unknownpopulation means of observed random variables in addition to structuralparameters. Inference consists of finding confidence intervals for functions ofthe structural parameters. Our theory provides finite-sample lower bounds onthe coverage probabilities of the confidence intervals under three sets ofassumptions of increasing strength. With the moderate sample sizes found inmost economics applications, the bounds become tighter as the assumptionsstrengthen. We discuss estimation of population parameters that the boundsdepend on and contrast our methods with alternative methods for obtainingconfidence intervals for partially identified parameters. The results of MonteCarlo experiments and empirical examples illustrate the usefulness of ourmethod.

Inference in a class of optimization problems: Confidence regions and finite sample bounds on errors in coverage probabilities