Consistent Estimation of Shape-Restricted Functions and Their Derivatives

We examine the estimation problem for shape-restricted functions that are continuous, non-negative, monotone non-decreasing, and strictly concave. A sieve estimator based on bivariate Bernstein polynomials is proposed. This estimator is drawn from a sieve, a set of shape-restricted Bernstein polynomials, which grows with the sample size in such a way that it becomes dense in the set of shape-restricted functions. Under some mild conditions, we show that this sieve estimator of the true function and the estimators of its first and second derivatives are uniformly consisten. THe estimators of elasticities of substitution are uniformly consistent as well.