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

## TL;DR

This paper introduces three non-asymptotic methods for inference on partially identified parameters in optimization problems, providing finite-sample coverage bounds and demonstrating their effectiveness through simulations and empirical data.

## Contribution

It develops finite-sample bounds for confidence intervals in partially identified models, extending inference methods to a broader class of optimization-based problems.

## Key findings

- Finite-sample bounds improve with stronger assumptions.
- Monte Carlo experiments validate the methods.
- Empirical examples demonstrate practical usefulness.

## Abstract

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.