Estimation And Tests In Linear Mixed Models

In this paper, we study the estimation and tests of unknown parameters in the linear mixed model. Also some generalized p-value solutions for multivariate Behrens-Fisher problem are proposed.Panel data models are special mixed models which often appear in repeated experiments, multistage sampling surveys and economic surveys with unit effects and time effects. The panel data models are widely used in econometrics, market researches and regional economic surveys, etc.In the panel data models, testing the regression coefficient is need in the variable selections and judgements of the validity of the linearity of the models. When the variance components are known, there are uniformly powerful tests. But in actual fact, they are often unknown and are substituted with their estimates. Different test statistics come from different estimating methods, and their test sizes and power functions are unknown because of their unknown distributions. In this paper, we use the method of generalized p value to construct exact tests. Simulation show that the tests are more, powerful than other tests with testing size approximating nominal level. Also, generalized confidence spheres of regression coefficient are proposed with the concept of generalized confidence region.For the panel data models, many tests were proposed to test whether the variances of the random effects are zeros. But for testing the hypothesis whether the variances are smaller than a specified nonnegative value, they all do not work well. In this paper, two tests are proposed to test the hypothesis with generalized p values and QR decomposition of design matrices. Also a generalized confidence interval for the variance component is proposed.Inference on the mean vectors of several multivariate normal populations often appears around us, such as quality tests and the controls. It is very difficult to test the mean vectors when the covariance matrices of the populations are different* The substitution of the covariance matrices with their estimates often results in the unknown distributions of the test statistics. Bias appears when the differences are ignored. In this paper, we get the distribution of the Bartlett's decomposition of sample covariance matrices. By this distribution, exact tests are constructed with the concept of generalized p values. Simulation show these tests are more powerful than existed tests. Also, a traditional test method is proposed by the Bartlett's decomposition of sample covariance matrices and the sample mean vectors. Simulation show that the type I error probability of the test is slightly smaller than the nominal level.Linear mixed models are linear models which are widely used in econometrics, biology, and medicine, etc. The analysis of variance (ANOVA) estimate is an important method to estimate the variance components in the linear mixed models. In the linear mixed model with three variance components, the ANOVA estimate is improved in the sense of mean-squared errors, and the result is generalized to the general linear mixed model. ANOVA estimate is not a nonnegative estimate, so it is very interesting to construct positive estimates of variance components. In the mixed linear model with two variance components, two positive estimates of variance component are proposed, which are dominate the ANOVA estimate and the Tatsuya estimate in the sense of mean-squared errors. Also positive estimates of variance components are proposed, which have smaller mean-square error than ANOVA estimates, in two way classification models with random effects.Restricted maximum likelihood estimation is one of the most important estimate methods in the mixed linear models. But in the most cases, iterative algorithms, such as EM algorithm, must be used. The QR decomposition on design matrices transforms the design matrices into upper-triangle matrices, then the orders of the matrices used in the iterative process can be decreased and the amount of data is reduced. Simulation show that the QR decomposition can make EM algorithm run much more quickly and the almost same results are got whether QR decomposition are used or not. Also, a new algorithm for ANOVA estimate is proposed with QR decomposition.