论文标题:线性混合效应模型协方差阵的估计问题
Covariance Matrix Estimate in Linear Mixed Models
论文作者
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 742,论文页数 50页File Size961K
2006-05-01论文网 http://www.lw23.com/lunwen_316532/
Least risk estimate ;The spectral decomposition estimate ;Loss function;Risk function;Mean squared error
本论文是对线性混合效应模型中参数的谱分解估计方法的深入讨论。我们知道，由谱分解方法得到的参数的估计有很多优良的性质。其中，对于观测向量协方差阵的谱分解估计，我们很容易得到它在一些损失下的风险函数。本文就是基于观测向量协方差阵的谱分解估计的这一性质展开讨论的。 首先，我们基于观测向量协方差阵的谱分解估计，提出了观测向量协方差阵的一类估计(谱分解估计的加权形式)，然后求其在Stein损失和Etropy损失下的风险函数，目的是讨论这类估计中的一致最优估计(风险最小)，从而得到观测向量协方差阵的新估计。在Etropy损失下得到的观测向量协方差阵的新估计同其谱分解估计相同；在Stein损失下得到了异于其谱分解估计的新估计。同时，我们证得：在一些模型中，在Etropy损失下，新估计的风险函数比其谱分解估计，ANOVAE和MINQUE的风险函数小。 其次，我们给出了观测向量协方差阵特征根及方差分量的压缩估计，并证明了这些新估计的一些统计性质；另外，我们还证明了观测向量协方差阵的特征根及方差分量的压缩估计在均方误差意义下优于其谱分解估计。 最后，我们把上述结果应用到了具体模型中，再次验证了我们的结果，并且，我们证明了在单向和两项分类模型中方差分量的压缩估计取负值的概率要比其谱分解估计取负值的概率小。
The thesis is concerned with the further argumentation of the spectral decomposition estimation in linear mixed models. It is known that the spectral decomposition estimate has many properties.Thereinto, for the spectral decomposition estimate of the covariance matrix ,we can gain the risk functions under some losses.The thesis is based on the above-mentioned property.The discussions are as follows:Firstly,we propose new kind of estimates of covariance matrix based on its spectral decomposition estimate (its weighted type),and obtain their corresponding risk functions under Stein loss and Ertopy loss.Thereby we discuss the best estimate in these new estimates in virtue of the functions.Furthermore we gain the new estimation of the covariance (the weighted type of its SDE) and call the reduced es-timate.Under Etropy loss, this new estimate of the covariance matrix is its spectral decomposition estimate.But its new estimate is differ from the spectral decomposition estimate of the covariance matrix. At the same time,we prove the property that the risk function of the new estimate of the covariance matrix is less than its spectral decomposition estimate,ANOVAE and MINQUE under Stein loss in some models.Secondly,we give the reduced estimates of the latent roots and the variance components and prove some properties of these reduced estimates. In addition,it is proved that the reduced estimates of the latent roots and the variance components are superrior to their spectral decomposition estimates under the meaning of MSE.
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