Shape-Constrained Density Estimation via Optimal Transport

Constraining the maximum likelihood density estimator to satisfy a sufficiently strong constraint, $\log-$concavity being a common example, has the effect of restoring consistency without requiring additional parameters. Since many results in economics require densities to satisfy a regularity condition, these estimators are also attractive for the structural estimation of economic models. In all of the examples of regularity conditions provided by Bagnoli and Bergstrom (2005) and Ewerhart (2013), $\log-$concavity is sufficient to ensure that the density satisfies the required conditions. However, in many cases $\log-$concavity is far from necessary, and it has the unfortunate side effect of ruling out sub-exponential tail behavior. In this paper, we use optimal transport to formulate a shape constrained density estimator. We initially describe the estimator using a $\rho-$concavity constraint. In this setting we provide results on consistency, asymptotic distribution, convexity of the optimization problem defining the estimator, and formulate a test for the null hypothesis that the population density satisfies a shape constraint. Afterward, we provide sufficient conditions for these results to hold using an arbitrary shape constraint. This generalization is used to explore whether the California Department of Transportation's decision to award construction contracts with the use of a first price auction is cost minimizing. We estimate the marginal costs of construction firms subject to Myerson's (1981) regularity condition, which is a requirement for the first price reverse auction to be cost minimizing. The proposed test fails to reject that the regularity condition is satisfied.