Inference On Variance Components Under Linear Mixed Models

Linear mixed effects model is one of the most widely used model in modern statistics.Linear mixed models fit not only average but also the covariance matrix structures through decomposing the random part of models into random effects and errors, so it is widely applied in the analysis of repeated measurement data such as genetic data, longitudinal data, panel data.It is the first thing to estimate the variance components in studying linear mixed models, and research on it holds an important position in linear mixed effects model history. This dissertation mainly considers the statistical inference on variance components under linear mixed models,and studies the estimation and exact tests on variance components under normal errors and skew-normal errors, respectively.In Chapter 2, we compare two estimators of the variance components in general linear mixed models, the ANOVA estimator and the spectral decomposition(SD) estimator. Based on the spectral decomposition of covariance matrix, two sufficient conditions for the identity of ANOVA estimator and SD estimator were established. Furthermore, under normal setting, we prove that the two estimators are not only identical, but also the uniformly minimum variance unbiased estimator.Chapter 3 considers linear mixed models with skew-normal errors, and prove that the ANOVA-type estimator of variance components still is unbiased; F and Fβstatistic were proposed for the hypotheses on the random effects and fixed effects, respectively; which both follow F distribution when the corresponding null hypothesis is true. At last some numerical simulations are given to illustrate our results.