Estimating Partial Linear Regression with Shape Constraints Using Second-order Least Squares Estimation Approach

This article proposes a semi-parametric approach to partial linear models for estimations of unknown nonparametric components with shape constraints and parametric components. We use Bernstein polynomials to approximate the unknown regression function of geometric restrictions and apply the second-order least squares method to compute the estimators collectively. Advantages of our approach are as follows: for one thing, random errors are not necessarily to be parametric settings except for finite fourth moments assumptions; for another, supposed properties of geometric restrictions of the nonparametric component, if applicable, have a strong stabilizing effect on the estimates. We use bootstrap methods to find the optimal choice of dimensions of Bernstein estimators. The methods are illustrated using a series of simulated data from partial linear models satisfying a variety of shape restrictions. In particular, we will examine the performance of regression estimates in the analysis of an air pollution data.