论文标题:一类混合效应模型的参数估计
Parameters Estimation in Mixed-Effect Models
论文作者 尹素菊
论文导师 王松桂,论文学位 硕士,论文专业 概率论与数理统计
论文单位 北京工业大学,点击次数 128,论文页数 47页File Size1096k
2001-05-01论文网 http://www.lw23.com/lunwen_1007109187/ ANOVA,方差分量,两步估计,谱分解,混合线性模型, 平衡模型,无偏估计,变换不变量,均方误差
AVOVA, spectral decomposition, two-stage estimator, mixed linear model, balance data model, unbiased estimator, translation invariance statistics, mean squares error
由于线性混合模型在生物、医学、经济、计算机、微波工程等领域具有十分广泛的应用，因此，对这种模型的统计研究颇受统计学家的重视。对方差分量的估计统计学家们提出了许多方法，如方差分析估计，极大似然估计，限制极大似然估计，最小范数二次无偏估计等。这些方法都是把固定效应和方差分量的估计分开来进行的，除了方差分析法外，它们都需要解一个非线性方程组，一般都没有显式解，只能获得迭代解。本文基于模型协方差阵的谱分解，提出了一种新的估计方法。这种方法有一些优点，（1）可以获得固定效应的若干个可行估计，即不包含方差分量的估计，在一些问题中（见后面例1.3.1），这些估计本身有良好的性质，便于应用。（2）用这些估计的线性组合可以获得固定效应的广义最小二乘估计。（3）通过固定效应的可行估计获得方差分量的估计。它们只需要解一个线性方程组。我们的研究结果表明，在一些特殊模型中，特别是只包含一个方差分量的线性模型，所获得的固定效应和方差分量的估计都是具有良好的统计性质，至于含多个方差分量的线性模型，新估计的统计性质有待进一步深入研究。应用新的估计方法我们计算了一些常见的平衡模型，我们的结论是，在随机模型中，与方差分析法得出的方差分量的估计是相同的，而在混合平衡模型中，只有无交互效应及有交互效应的两向分类模型中，两种估计方法得出的方差分量的估计不相同。对于一般线性模型，本文还给出了回归系数线性组合的两步估计的均方误差的表达式。
Because the mixed-linear model is important in biology, medicines, economics, micro- computer, microwave engineering and other fields, so that the statisticians take more attention to the statistical research in this model. Concerning the estimation of the variance components, some estimators, for example, the Analysis of Variance Estimator (ANOVAE), Maximum Like- lihood Estimator (MLE), Restricted MLE, the minirnutn norm quadratic unbiased estimator (MINQUE), and so on, have been proposed in the literature. For these estimators, computation of the estimates of the fixed effect and the variance components are separated, and except of the ANONAE, they need solve a non-linear equations, unluckily, which do not have explicit solution, and only have an iteration solution in general. In this paper, on the basis of the decomposition of the covariance matrix, we l)roposed a new method that it can estimate the variance components and their fixed effect parameters simultaneously. This method has the following superiority over other estimators. Firstly, we can get several estimators of the fixed effect parameters, and these estimators do not contain the unknown parameters, i.e. these estimators are the feasible estimator, further, in some problems (see in the following example 1.3.1), those estimators have some good statistical properties. Secondly, we can use of the linear combination of these feasible estimators to get the generalized least squares estimator (GLSE) of the fixed effect parameters. And thirdly, on the basis of every feasible estimator, we can also get the estimates of the van- amice components only by solving a linear equations. Our research show that, in some models, especially in the mixed models with one variance components, the estimates of the fixed effect and the variance component obtained by our new method have good statistical properties, as far as the linear models contain more variance components, the statistical properties of the new method needs study deeply. In the rest of the paper, we apply the new method to some balance models, for example, mixed and random effect models, and compare the result with AVOVA estimate, we can observe that in those models, the results of the random models is same with the AVOVA, but in the mixed-effect models, the two-way crossed classification with interaction or no interaction, we get a different result frommm AVOVA estimate. In this paper, we also consider the unbiasedness of the two-stage estimate on basis of tIme estimates of variance components with time new tmmethod. For the general linear models, we also obtain the expression of the mean squared error of the two-stage estimator of regression coeffients.
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