A difference-based method for testing no effect in nonparametric regression

The paper proposes a novel difference-based method for testing the hypothesis of no relationship between the dependent and independent variables. We construct three test statistics for nonparametric regression with Gaussian and non-Gaussian random errors. These test statistics have the standard normal as the asymptotic null distribution. Furthermore, we show that these tests can detect local alternatives that converge to the null hypothesis at a rate close to \(n^{-1/2}\) $n^{-1/2}$ previously achieved only by the residual-based tests. We also propose a permutation test as a flexible alternative. Our difference-based method does not require estimating the mean function or its first derivative, making it easy to implement and computationally efficient. Simulation results demonstrate that our new tests are more powerful than existing methods, especially when the sample size is small. The usefulness of the proposed tests is also illustrated using two real data examples.