Optimal transportation is used to define a nonparametric density estimator as the solution of a convex optimization problem. The framework allows for density estimation subject to a variety of shape constraints, including \rho-concavity and Myerson's (1981) regularity condition. The mean integrated squared error for the density estimator of a random variable in \mathbb{R the  California Department of Transportation to explore whether their choice of awarding construction contracts using a first price auction is cost minimizing  .
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 .
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Optimal transportation is used to define a nonparametric density estimator as the solution of a convex optimization problem. The framework allows for density estimation subject to a variety of shape constraints, including \rho-concavity and Myerson's (1981) regularity condition. The mean integrated squared error for the density estimator of a random variable in \mathbb{R
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
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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
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to explore whether their choice of awarding construction contracts using
's decision to award construction contracts with the use of
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. 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