Least Concavity and the Distribution-Free Estimation of Non-Parametric Concave Functions

This paper studies the estimation of fully nonparametric models in which we can not identify the values of a symmetric function that we seek to estimate. I develop a method of consistently estimating a representative of a concave and monotone nonparametric systematic function. This representative possesses the same isovalue sets as the systematic function. The method proceeds by characterizing each set of observationally equivalent concave functions by a unique "least concave" representative. The least concave representative of the equivalence class to which the systematic function belongs is estimated by maximizing a criterion function over a compact set of least concave functions. I develop a computational technique to evaluate the values, at the observed points, and the gradients, at every point and up to a constant, of this least concave estimator. The paper includes a detailed description of how the method can be used to estimate three popular microeconometric models.