论文标题:线性混合效应模型的估计与检验
Estimation and Tests in Linear Mixed Models
论文作者
论文导师 王松佳,论文学位 博士,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 167,论文页数 111页File Size1607K
2007-04-01论文网 http://www.lw23.com/lunwen_312438192/
Linear mixed model;Panel data; Generalized p values; Generalized confidence region; Behrens-Fisher problem; QR decomposition; Exact test; Vari-ance component
本文主要研究线性混合模型中未知参数的估计和检验问题。同时本文也给出了多元Behrens—Fisher问题的几种广义p值解。 Panel数据模型是一种线性混合模型，常常产生于重复测量试验，多级抽样调查以及含时间和个体效应的经济调查。在计量经济学、市场分析、区域经济调查等领域有着广泛地应用。 在panel数据模型中，对回归系数的检验，常常用于变量选择或检验模型线性假设的合理性。如果方差分量已知，则存在一致最优功效检验。方差分量未知时，常用它们的估计代替它们。不同的方差分量的估计，就得到不同的检验统计量。这些检验统计量的分布都是未知的，在小样本情况下，很难控制它们的检验水平和功效。本文采用广义p值的方法，给出了一种精确的检验。模拟结果显示，这种检验能很好的控制检验水平，并且有更高的检验功效。本文同时给出了回归系数的广义置信域。 存panel数据模型中，对方差分量是否为0的检验已经有了很多方法，但检验方差分量小于等于某个指定的常数，已有的检验很难使用。本文利用设计阵的QR分解和广义p值，给出了一种精确的检验。同时还给出方差分量的一个广义胃信区间。 对多个多元正态总体均值的检验是在生产实践和社会生活中经常遇到的一类检验问题，比如产品质量的检验和控制。如果正态总体的协方差矩阵是不同的，这类问题常常称为Behrens-Fisher问题。协方差矩阵的不同给检验问题带来巨大的困难。用它们的估计代替，则检验统计量分布未知。而把协方差矩阵当作相同的来处理，又会带来偏差。本文中我们给出了样本协方差矩阵Bartlett分解的分布，同时利用广义p值，给出了Behrens-Fisher问题的几种精确的检验方法。模拟结果表明，这些方法比已有的方法有更高的功效。另外本文还利用样本协方差矩阵的Bartlett分解和样本均值，给出了检验共同均值的一种传统方法，这种检验犯第一类错误的概率比标称的检验水平略低。 线性混合模型不仪对均值部分建立模型，而且对协方差矩阵建模。它能够处理更为复杂的问题，因此在生物学、计量经济学、医学等需要复杂建模的领域，得到越来越广泛的应用。在线性混合模型中，ANOVA估计是常用的估计方差分量的方法。对含三个方差分量的线性混合模型，本文在均方误差意义下，改进ANOVA估计，并把这一结果推广到一般的线性混合模型上。ANOVA估计得到的往往不是非负的，构造方差分量的非负估计，一直是令人感兴趣的问题。对含有两个方差分量的线性混合模型，本文给出了两个正的截尾估计，并指出它们在均方误差意义下优于ANOVA估计和Tatsuya估计。并把这一方法应用到两向分类随机效应模型，给出其中两个方差分量的正估计。 限制极大似然估计也是一种很重要的估计方差分量的方法，但是它常常需要通过迭代法求解。EM算法是其中一种重要的迭代算法。对设计阵的QR分解，可以把设计阵变换成上三角矩阵。这样可以降低参与迭代运算的矩阵的阶数，减少了参与运算的数据量，从而提高运算的速度。本文把QR分解应用到EM算法中，并用模拟的方法验证了QR分解可以极大的提高运算的速度。同时本文利用设计阵的QR分解，给出了ANOVA估计的一种新方法，在一般情况下，新方法更便于计算。
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.
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