The proximal bootstrap for constrained estimators

We demonstrate how to conduct uniformly asymptotically valid inference for $\sqrt{n}$-consistent estimators defined as the solution to a constrained optimization problem with a possibly nonsmooth or nonconvex sample objective function and a possibly nonconvex constraint set. We allow for the solution to the problem to be on the boundary of the constraint set or to drift towards the boundary of the constraint set as the sample size goes to infinity. We construct a confidence set by benchmarking a test statistic against critical values that can be obtained from a simple unconstrained quadratic programming problem. Monte Carlo simulations illustrate the uniformly correct coverage of our method in a boundary constrained maximum likelihood model, a boundary constrained nonsmooth GMM model, and a conditional logit model with capacity constraints.