The Robust Inference In Linear Mixed Model With Generalized Skew-normal Error

The linear mixed effects model is a kind of very important statistical model.It has unique advantages in dealing with repeated measurement data,block data,and spatially related data.In the literature,it is generally assumed that both random effects and errors follow the normal distributions.However,the normality assumption is too restrictive in most practical applications and almost all data sets,collected from many diverse fields,are skewed,such as systolic blood pressure data,cholesterol data,blood glucose data,triglycerides data eta.in the Framingham Heart Study.In this paper,we mainly study the robust inference of linear mixed model with generalized skew-normal distribution errors.In Chapter 2,we introduce the definition,quadratic,moment generating function,expectation and covariance of a class of generalized skew-normal(GNS)distribution.And we prove that the distribution of linear combination of a GSN vector and a normal vector still follows GSN distribution,present the expression of its density function and corresponding results under several special skewed distributions.In Chapters 3,we mainly study the parameter inference for the linear mixed model with GSN distribution error.We give ANOVA-type estimation of components of variances,ANOVAtype F-tests on random effect and fixed effect,and the corresponding confidence interval of fixed effect.Furthermore,it is proved under the GSN distribution assumption that the ANOVA-type estimates of variance components are unbiased,both the ANOVA-type F-tests on the significance of random effect and fixed effect are exact F-tests,the ANOVA-type inference is robust to the distribution of random effect.Simulation study shows that the power of the ANOVA-type test on on the significance of random effect is almost not affected by the skewed function.In Chapters 4,we applied the above method to the analysis of the Framingham heart study data.Since systolic blood pressure,cholesterol,and blood glucose vary with age,we fit these data by linear mixed models with GSN distribution error,respectively.Results of The ANOVAtype F-test show that individual random effects is significant for all three indicators and change over time.Further,we also make comparisons on the three indicators between individuals who suffer from hypertension and individuals who do not,and between individuals who have hypertension treatment and individuals who do not.