We describe and examine a test for  shape constraints, such as monotonicity, convexity (or both simultaneously) , U-shape, S-shape and others, in a nonparametric framework using partial sums empirical processes. We show that, after a suitable transformation, its asymptotic distribution is a functional of the standard Brownian motion, so that critical values are 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.
We describe and examine a test for a general class of shape constraints, such as constraints on the signs of derivatives, U-(S-)shape, symmetry, quasi-convexity , log-convexity, r-convexity, 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 the standard Brownian motion, so that critical values are 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.
A
a general class of
R
monotonicity, convexity (or both simultaneously)
constraints on the signs of derivatives, U-(S-)shape, symmetry, quasi-convexity
R
U-shape, S-shape and
log-convexity, r-convexity, among