Checking normality of model errors under additive distortion measurement errors

Abstract

We study the goodness-of-fit tests for checking the normality of the model errors under the additive distortion measurement error settings. Neither the response variable nor the covariates can be directly observed but are distorted in additive fashions by an observed confounding variable. The proposed test statistics is based on logarithmic transformed variables with residuals and a particular choice of the kth power covariance-based estimator. The proposed test statistics has three advantages. Firstly, the asymptotic null distribution of the test statistics are obtained with known asymptotic variance. Secondly, the test statistic tests are irrelevant to the model. Thirdly, the proposed test statistics automatically eliminate the additive distortion effects involved in the response and covariates. The simulation studies show the proposed test statistics can be used to check normality when the sample size is very large. A real example is analysed to illustrate its practical usage.

KEYWORDS:

• Calibration
• checking normality
• distribution-free tests
• additive distortions

MATHEMATICS SUBJECT CLASSIFICATION (2000):

• 62G05
• 62G08
• 62G20

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors thank the editor, the associate editor, and two referees for their constructive suggestions that helped us to improve the early manuscript. Mengyao Li's research was supported by the National Social Science Foundation of China [grant number 21BTJ048, Principal Investigator, 2021.06-2024.06]. Jun Zhang's research was supported by the National Natural Science Foundation of China [grant numbers 12371448 and 12071305]. Yan Zhou's research received support from the National Natural Science Foundation of China [grant numbers 12071305 and 12371295] and Natural Science Foundation of Guangdong Province of China [grant number 2023A1515011399]. Jiangshe Zhang's research was supported by the National Natural Science Foundation of China [grant number 12371512].