Convex analysis in the semiparametric model with Bernstein polynomials

In this paper, we propose Bernstein polynomial estimation for the partially linear model when the nonparametric component is subject to convex (or concave) constraint. We employ a nested sequence of Bernstein polynomials to approximate the convex (or concave) nonparametric function. Bernstein polynomial estimation can be obtained as a solution of a constrained least squares method and hence we use a quadratic programming algorithm to compute efficiently the estimator. We show that the estimator of the parametric part is asymptotically normal. The rate of convergence of the nonparametric function estimator is established under very mild conditions. The small sample properties of our estimation are provided via simulation study and compared with regression splines method. A real data analysis is conducted to illustrate the application of the proposed method.