Checking normality of model errors under additive distortion measurement errors

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.