Single-index relative error regression models

Abstract

We consider the estimation and hypothesis test for single-index model by proposing a profile mean squares relative error estimation method, coupling with a leave-one-component-out method. The asymptotic normality for the PMSRE estimator is studied. For the hypothesis test of the single-index parameter, a restricted estimator under the null hypothesis and test statistics are proposed. The asymptotic properties for the restricted estimator and test statistic are established. Lastly, we make some comparisons between the proposed estimator and the profile least squares estimator and the profile least product relative estimator through the simulation. Our simulation studies indicate that the profile mean squares relative error estimator provides a better alternative under some settings, particularly for multimodal or asymmetric distributions of the error term.

Keywords:

• Profile mean squares relative error estimator
• Restricted estimator
• Single-index

MATHEMATICS SUBJECT CLASSIFICATION (2000):

• 62G05
• 62G08
• 62G20

Acknowledgments

The authors thank the editor, the associate editor, and two referees for their constructive suggestions that helped us to improve the early manuscript.

Additional information

Funding

Yujie Gai’s research was supported by the National Natural Science Foundation of China (Grant No. 12071497), the National Statistical Science Research Project (Grant No. 2020LY014), the Emerging Interdisciplinary Project of CUFE and the Discipline Construction Funds of Central University of Finance and Economics. Jun Zhang’s research was supported by the Natural Science Foundation of Guangdong Province (Grant No. 2020A1515010372), and the University stability support program A of Shenzhen (Grant No. 20200813151828003).