Principled nonparametric tests for regression curvature in Rd
are often statistically and computationally challenging. This paper introduces
the stratified incomplete local simplex (SILS) tests for joint concavity of
nonparametric multiple regression. The SILS tests with suitable bootstrap
calibration are shown to achieve simultaneous guarantees on dimension-free
computational complexity, polynomial decay of the uniform error-in-size, and
power consistency for general (global and local) alternatives. To establish
these results, a general theory for incomplete U-processes with stratified
random sparse weights is developed. Novel technical ingredients include maximal
inequalities for the supremum of multiple incomplete U-processes