Testing For Normality In Linear Mixed Models

Longitudinal data often appears in the biological, economic, weather, industry and other fields. In the study of the continuous longitudinal data, the ordinary linear regression model is obviously not a good model, We usually use linear mixed-effects models to make statistical analysis. The random errors and random effects are often assumed to obey normal distribution, then we can use maximum likelihood estimation (MLE) or restricted maximum likelihood estimation (RMLE) to estimate fixed effects and random effects, study the asymptotic properties easily, and get good results. However, in fact, this assumption is not always true. This paper mainly studies how to test the normality of linear mixed-effects models and how to select and estimate fixed effects.Since the random errors are unobservable, we need to estimate random effects and fixed effects before the normality test. In this paper, we use QR decomposition method to remove the random effects. On this basis, we use SCAD (Smoothly clipped absoluted deviation) method to select and estimate the fixed effects. Theoretical studies have shown that the estimator is (?)-consistent under some regular conditions. Then we extend the BHEP (Baringhaus-Henze-Epps-Pulley) test to construct our test statistic based on the estimation of random errors. In the study of the asymptotic properties, the test statistic asymptotically converges to a zero-mean Gaussian process under the null hypothesis. Some simulation studies are applied to verify the effectiveness of our method.