Non-asymptotic inference in a class of optimization problems

This paper describes a method for carrying out non-asymptotic inference on partially identified parameters that are solutions to a class of optimization problems. The optimization problems arise in applications in which grouped data are used for estimation of a model's structural parameters. The parameters are characterized by restrictions that involve the population means of observed random variables in addition to the structural parameters of interest. Inference consists of finding confidence intervals for the structural parameters. Our method is non-asymptotic in the sense that it provides a finite-sample bound on the difference between the true and nominal probabilities with which a confidence interval contains the true but unknown value of a parameter. We contrast our method with an alternative non-asymptotic method based on the median-of-means estimator of Minsker (2015). The results of Monte Carlo experiments and an empirical example illustrate the usefulness of our method.