论文标题:线性混合模型参数估计问题的研究
Research on the Parameter Estimation of the Mixed-Linear Models
论文作者
论文导师 郭鹏江,论文学位 硕士,论文专业 应用数学
论文单位 西北大学,点击次数 113,论文页数 50页File Size1197k
2005-05-01论文网 http://www.lw23.com/lunwen_399809062/
Linear mixed models; Variance component; Spectral decomposition; Empirical Bayes estimate; Asymptotical optimality; Convergence rate; Two-stage predictor; Mean square 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。这种模型的未知参数分两类,一类是固定效应,一类是方差分量。关于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,他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。 近年来,王松桂等提出了固定效应和方差分量的一种新估计,称为谱分解估计。新估计的突出特点是,固定效应的估计是具有良好统计性质的线性估计。本文第二章对这种估计的方法、性质和应用进行了系统的介绍和总结,并提出今后的研究方向。此外还对当前研究的一些热点问题以及关于线性混合模型的最新研究成果作了适当的介绍和说明。 第三章对含两个方差分量的线性混合模型在加权平方损失下导出方差分量的Bayes估计,利用多元密度及其偏导数的核估计方法构造了方差分量的经验Bayes估计,并建立了这个估计的收敛速度。 最后,对一般的线性混合模型,本文研究其固定效应和随机效应线性组合的两步预测问题,给出了两步预测均方误差的一个近似计算公式,为比较不同方差分量估计的优劣性进一步奠定了基础。
Because of the wide usage of mixed-linear model in biology, medicine, computer and microwave engineering and other fields, the statistic research on this model received increasing attention. The unknown parameters of this model are divided into two categories: one is the fixed effects, and the other is the variance components. With regard to the estimation of the variance components, many statisticians proposed a number of approaches, such as the Analysis of Variance Estimator, Maximum Likelihood Estimator, Restricted MLE, and the Minimum Norm Quadratic Unbiased Estimator, etc. For these estimators, computation of the estimators of the fixed effects and the variance components are separated. Except the Analysis of Variance Estimator, these approaches all need to solve a non-linear equation, which does not have explicit solution, and only has an iteration solution in general.Recently, Wang Songgui and some other people put forward a new estimation approach for the fixed effects and variance components, which is called spectral decomposition estimator. The peculiarity of this new approach is that the estimation for the fixed effects is the linear estimation which possesses a good quality for estimation. The second chapter of the present thesis attempts to make a systematic introduction and generalization on its methods, qualities and applications and propose the future research direction. In addition, some hot issues concerning the current researches and the newest research results about the mixed-linear models are being appropriately discussed and explained.In chapter three, Bayes estimators of variance components are derived for mixed-linear models with two variance components under the weighted square loss function, and the empirical Bayes estimators are constructed by the kernel estimation method of multivariate density and its mixed partial derivatives, and the convergence rates of this estimator are established.Finally, as far as the general mixed-linear model is concerned, the present thesis explores the two-staged prediction of the linear combination about the fixed effects and the random effects and meanwhile an approximate computation formula is derived for the mean square error of the two-staged prediction which paves the way for comparing and contrasting the advantages and disadvantages of different variance components.
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