线性混合模型中参数估计的研究

发布时间：2018-07-05 08:50

本文选题：线性混合模型 + 固定效应　；参考：《青岛科技大学》2015年硕士论文

【摘要】：线性混合模型在生物、经济、计算机等领域具有很广泛的应用,其参数估计问题是统计学家研究的重点之一。本文针对固定效应与随机效应两类参数的估计方法进行研究,并得出一些新的结论。对于固定效应的参数估计,介绍了线性可估函数c'β的最小二乘估计(LSE)与两步估计分别为最佳线性无偏估计(BLUE)的条件,并分析了BLUE的可容许性。在此基础上,将线性混合模型转化为满足Gauss-Markov段设的模型,给出在该假设下LSE为BLUE的充分条件；探讨了利用奇异值分解对两步估计进行改进的方法,并讨论了改进后两步估计具有的一些新性质；由于LSE经常会有精度上的损失,引进相对效率的概念,以生长曲线模型与权回归模型为例,分别定义了两种模型新的相对效率,并给出它们的上下界。新的相对效率考虑了各分量之间协方差产生的影响,提高了灵敏度。对于随机效应的参数估计,分别介绍了方差分析估计(ANOVAE)与谱分解估计(SDE)方法的相关性质以及具体应用。在一个新的估计类中,改进了ANOVAE方法,探讨了改进估计的非负性,同时,基于全空间的多层正交直和分解,将ANOVAE方法推广到不依赖于随机效应正态假设的线性混合模型中；进一步研究SDE的性质,将SDE推广到一般的线性混合模型中,讨论了SDE与ANOVAE相等的充要条件,给出三个具体实例进行分析验证。
[Abstract]:Linear mixed model has been widely used in biology, economy, computer and so on. The parameter estimation problem is one of the key points in the research of statisticians. In this paper, two kinds of parameter estimation methods, fixed effect and random effect, are studied, and some new conclusions are obtained. For the parameter estimation of fixed effect, the condition that the least square estimator (LSE) and the two-step estimator of the linear estimable function c'尾 are respectively the best linear unbiased estimator (blue) are introduced, and the admissibility of blue is analyzed. On this basis, the linear mixed model is transformed into a model satisfying Gauss-Markov segment, and the sufficient conditions for LSE to be blue under this assumption are given, and the method of improving the two-step estimation by using singular value decomposition is discussed. Some new properties of the improved two-step estimation are discussed. As LSE often has a loss of precision, the concept of relative efficiency is introduced. Taking growth curve model and weight regression model as examples, the new relative efficiency of two models is defined respectively. Their upper and lower bounds are given. The new relative efficiency takes into account the influence of covariance between components and improves the sensitivity. For the parameter estimation of random effects, the properties and applications of ANOVAE and SDE are introduced. In a new class of estimators, the ANOVAE method is improved, and the nonnegativity of the improved estimation is discussed. At the same time, the ANOVAE method is extended to the linear mixed model which does not depend on the assumption of random effect normality based on the multilayer orthogonal sum decomposition in the whole space. The properties of SDE are further studied, and the SDE is extended to the general linear mixed model. The sufficient and necessary conditions for the equality of SDE and ANOVAE are discussed, and three examples are given for analysis and verification.
【学位授予单位】：青岛科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O212.1

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