Testing symmetry of model errors for non linear multiplicative distortion measurement error models

Abstract

To study the symmetry and asymmetry of the model error under multiplicative distortion measurement errors setting, we propose a correlation coefficient-based measure between the distribution function and the root of density function. The unknown distribution function and density function are estimated from four kinds of residuals: the conditional mean calibration-based residuals, the conditional absolute mean calibration-based residuals, the conditional variance calibration-based residuals, and the conditional absolute logarithmic calibration-based residuals. We study the asymptotic results of the estimators of correlation coefficient-based measure under four calibrations. Next, we consider statistical inference of the correlation coefficient-based measure by using the empirical likelihood method. The empirical likelihood statistics are shown to be an asymptotically standard chi-squared distribution. Simulation studies demonstrate the performance of the proposed estimators and test statistics. A real example is analyzed to illustrate its practical usage.

KEYWORDS:

• Calibration
• local linear smoothing
• multiplicative distortion
• measurement errors
• correlation coefficient

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. Yue Zhou (ID: 2020193040) is a junior student majoring in Statistics at Shenzhen University. This work was done when the third author was supervised by the first author.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Jun Zhang’s research was supported by the Science and Technology Planning Project of Shenzhen Municipality, P.R.China (Grant No. 20220810155530001) and the National Natural Science Foundation of China (Grant No.12071305). Zhenghui Feng’s research was supported by Guangdong Basic and Applied Basic Research Foundation (Grant No. 2023A1515010884).