Pairwise distance-based heteroscedasticity test for regressions

In this study, we propose nonparametric testing for heteroscedasticity in nonlinear regression models based on pairwise distances between points in a sample. The test statistic can be formulated such that U-statistic theory can be applied to it. Although the limiting null distribution of the statistic is complicated, we can derive a computationally feasible bootstrap approximation for such a distribution; the validity of the introduced bootstrap algorithm is proven. The test can detect any local alternatives that are different from the null at a nearly optimal rate in hypothesis testing. The convergence rate of this test statistic does not depend on the dimension of the covariates, which significantly alleviates the impact of dimensionality. We provide three simulation studies and a real-data example to evaluate the performance of the test and demonstrate its applications.