Optimal Global Test for Functional Regression

TL;DR

This paper develops a minimax optimal generalized likelihood ratio test for the nullity of the slope function in functional linear models, achieving the best possible detection rate based on smoothing spline and covariance kernels.

Contribution

It introduces a new test that attains the minimax optimal rate for detecting non-zero slope functions in functional regression models.

Findings

The test achieves the minimax optimal detection rate.

Simulation results confirm finite-sample effectiveness.

Application to environmental data demonstrates practical utility.

Abstract

This paper studies the optimal testing for the nullity of the slope function in the functional linear model using smoothing splines. We propose a generalized likelihood ratio test based on an easily implementable data-driven estimate. The quality of the test is measured by the minimal distance between the null and the alternative set that still allows a possible test. The lower bound of the minimax decay rate of this distance is derived, and test with a distance that decays faster than the lower bound would be impossible. We show that the minimax optimal rate is jointly determined by the smoothing spline kernel and the covariance kernel. It is shown that our test attains this optimal rate. Simulations are carried out to confirm the finite-sample performance of our test as well as to illustrate the theoretical results. Finally, we apply our test to study the effect of the trajectories of oxides of nitrogen ($\text{NO}_{\text{x}}$) on the level of ozone ($\text{O}_3$).