Stratified incomplete local simplex tests for curvature of nonparametric multiple regression

Abstract

Principled nonparametric tests for regression curvature in ${\mathbb{R}}^d$ 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.

Accepted to Bernoulli

References in corpus (6)

• On local $U$-statistic processes and the estimation of densities of functions of several sample variables
• Approximating high-dimensional infinite-order $U$-statistics: statistical and computational guarantees
• Optimality of Maximum Likelihood for Log-Concave Density Estimation and Bounded Convex Regression
• A Projection Framework for Testing Shape Restrictions That Form Convex Cones
• Testing nonparametric shape restrictions
• Stratified incomplete local simplex tests for curvature of nonparametric multiple regression

Cited by in corpus (2)

• A Projection Framework for Testing Shape Restrictions That Form Convex Cones
• Stratified incomplete local simplex tests for curvature of nonparametric multiple regression