Test shape constraints in semiparametric model with Bernstein polynomials

In this article, we mainly discuss hypothesis tests to assess whether nonparametric function satisfies monotonicity or convexity in semiparametric partially linear model. For this purpose, we propose a new test statistic for monotonicity or convexity of nonparametric function based on Bernstein polynomials. Furthermore, we employ likelihood ratio statistic to test the significance of regression parameters of the model. We discuss the asymptotic properties of the tests for both nonparametric function and regression parameters. A simulation study is conducted to evaluate the finite sample performance compared with the other methods for shape tests. The method is illustrated by the fuel efficiency study.