Subgroup Testing in Threshold Quantile Regression Model

Detecting heterogeneity across latent subgroups is essential for understanding complex data structures in fields such as economics, medicine and social sciences. Threshold quantile regression (TQR) with change‐plane structures provides a flexible framework for such heterogeneity, especially in the presence of heavy‐tailed errors. However, a key challenge in subgroup detection lies in the nonidentifiability of threshold parameters under the null hypothesis, which invalidates classical testing approaches. To address this, we propose a difference of quantile loss test (DQLT) that constructs a supremum‐type test statistic based on differences in quantile loss functions. We derive the asymptotic distributions of the proposed test statistic under both the null and local alternative hypotheses and a bootstrap procedure is introduced to compute ‐values, effectively. The performance of our method is evaluated through extensive simulation studies, demonstrating well‐controlled type‐I error under the null hypothesis and enhanced power under the alternative hypotheses. We further apply the DQLT method to CHARLS 2015 and Boston housing datasets, illustrating its effectiveness in identifying subgroups with heterogeneous covariate effects.