A New Graphical Device and Related Tests for the Shape of Non-parametric Regression Function

We consider a non-parametric regression model $y = m(x) + \epsilon$ and propose a novel graphical device to check whether the $r$-th ($r \geqslant 1$) derivative of the regression function $m(x)$ is positive or otherwise. Since the shape of the regression function can be completely characterized by its derivatives, the graphical device can correctly identify the shape of the regression function. The proposed device includes the check for monotonicity and convexity of the function as special cases. We also present an example to elucidate the practical utility of the graphical device. In addition, we employ the graphical device to formulate a class of test statistics and derive its asymptotic distribution. The tests are exhibited in various simulated and real data examples.