Approximation and consistent estimation of shape -restricted functions and their derivatives

In this thesis we propose a functional form for the study of shape-restricted functions based on Bernstein polynomials. In approximation, sequences of Bernstein polynomials and their associated derivatives converge uniformly to a known function and its derivatives. This shape-preserving property allows Bernstein polynomials to provide global approximation for shape-restricted functions. To estimate shape-restricted functions, we introduce a sieve estimator that is based on Bernstein polynomials. We show that, under some mild assumptions, this sieve estimator and its first and second derivatives are uniformly consistent estimators of the true function and its corresponding derivatives. A uniformly consistent estimator of the elasticity of substitution is thus obtained. All of these estimators are straightforward to implement in an applied setting.