Inference in a class of optimization problems: Con?dence regions and ?nite sample bounds on errors in coverage probabilities

This paper describes three methods for carrying out non-asymptotic inference on partially identi?ed 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 identi?ed parameters are characterized by restrictions that involve the unknown population means of observed random variables in addition to the structural parameters of interest. Inference consists of ?nding con?dence intervals for the structural parameters. Our theory provides ?nite-sample lower bounds on the coverage probabilities of the con?dence 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 con?dence intervals for partially identi?ed parameters. The results of Monte Carlo experiments and empirical examples illustrate the usefulness of our method.