Testing Nonparametric Shape Restrictions

We describe and examine a test for a general class of shape constraints, such as signs of derivatives, U-shape, quasi-convexity, logconvexity, among others, in a nonparametric framework using partial sums empirical processes. We show that, after a suitable transformation, its asymptotic distribution is a functional of a Brownian motion index by the c.d.f. of the regressor. As a result, the test is distribution free and critical values are readily available. However, due to the possible poor approximation of the asymptotic critical values to the finite sample ones, we also describe a valid bootstrap algorithm.