| Linear mixed models have a wide range of applications in the fields of biology, economics, computers, etc., whose parameter estimation problem is one of the focus in the study of statisticians. This paper studies estimation methods for fixed effects and random effects of two types of parameters, and draw some new conclusions.For parameter estimation of fixed effects, this paper has introduced the LSE and the two-step estimation of the linear estimable function c’β respectively as the best linear unbiased estimation(BLUE), and analyzed the admissibility of the BLUE. On this basis, the linear mixed model is transformed to meet the Gauss-Markov assumption, under the hypothesis, the sufficient condition of LSE as BLUE is given; The singular value decomposition is used to improve the two-step estimation method, and discusses some new properties of the improved two-step estimation; Because there is often a loss of accuracy for LSE, it introduces the concept of the relative efficiency in growth curve model and weighted regression model, and define separately new relative efficiencies for two models, the upper and lower bounds of them are given too. New relative efficiencies considers the impact of the covariance between each component and improves the sensitivity.For parameter estimation of random effects, this paper has introduced specific application and related properties of ANOVAE and SDE. It has established a new estimate class, improved ANOVAE and discussed the non-negative of the improved estimate, at the same time, based on the multi-level orthogonal straight and decomposition of the space, it extends the ANOVAE method to the linear mixed model whose random effects is not depend on the normality assumption; Furthermore, it analysis the nature of the SDE deeply, the SDE is generalized to the general linear mixed model, and it discusses the sufficient and necessary condition that the ANOVA estimate is equal to the SDE, while three specific examples are given to verify the analysis. |
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