A general nonparametric framework for testing hypotheses about a class of function-valued parameters

Music City Center 

Performing inference on function-valued parameters, such as the regression function or the conditional average treatment effect (CATE), poses fundamental challenges in nonparametric models. For a class of smooth function-valued parameters that can be expressed as functions of a conditional distribution, we develop a nonparametric test to assess whether the function-valued parameter is constant. The test statistic is based on the norm of the difference between two parameters. We propose a near-estimator of the norm that attains a tractable limiting distribution under the null, when the norm is zero. Our method improves upon many existing approaches for estimating norms which exhibit poor asymptotic behavior under the null. As an illustration of our framework, we present three concrete applications: (1) testing null variable significance in regression; (2) testing constant conditional covariance; and (3) testing constant CATE. Simulation studies demonstrate strong performance, and we further apply the method to identify predictive biomarkers for adjuvant chemotherapy response in HER2-positive breast cancer patients.

Pathwise differentiability

Function-valued parameters

Equality of functionals

Hypothesis testing 

Section on Nonparametric Statistics