Adaptive Nonparametric Comparison of Regression Curves

Abstract

We propose a new test for comparison of two regression curves, which integrates generalized likelihood ratio (GLR) statistics (Fan et al., 2001) with the data-driven criterion of selecting the smoothing parameter proposed by Guerre and Lavergne (2005). The local linear nonparametric estimator is used to construct the GLR statistic. We prove that the corresponding test statistic is asymptotically normal and free of nuisance parameters and covariate designs under the null hypothesis. The test adapts to the unknown smoothness of the difference between two regression functions and can detect local alternatives converging to the null hypothesis at rate . The wild bootstrap technique is used to approximate the critical values of the test for small samples. A simulation study is conducted to investigate the finite sample properties of the new adaptive test and to compare it with some other available procedures in the literature. The simulation results demonstrate the sensitivity and robustness of the proposed approach.

Keywords:

• Comparison of two regression curves
• Data-driven criterion
• Generalized likelihood ratio
• Local linear smoother
• Wild bootstrap

Mathematics Subject Classification:

• Primary 62G07
• Secondary 62G09

Acknowledgments

The authors thank the Editor and an anonymous referee for many constructive comments that greatly improved the article. This work was supported by the NSF of Tianjin grant No. 07JCYBJC04300, the NNSF of China grant No. 10571093, 10771107, 10671099, 10711120448, and the SRFDP of China grant No. 20050055038.