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# 统计代写|广义线性模型代写Generalized linear model代考|Empirical BLUP

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## 统计代写|广义线性模型代写Generalized linear model代考|Empirical BLUP

In practice, the fixed effects and variance components are typically unknown. Therefore, in most cases neither the best predictor nor the BLUP is computable, even though they are known to be the best in their respective senses. In such cases, it is customary to replace the vector of variance components, $\theta$, which is involved in the expression of BLUP, by a consistent estimator, $\hat{\theta}$. The resulting predictor is often called empirical BLUP, or EBLUP.

Kackar and Harville (1981) showed that, if $\hat{\theta}$ is an even and translation-invariant estimator and the data are normal, the EBLUP remains unbiased. An estimator $\hat{\theta}=\hat{\theta}(y)$ is even if $\hat{\theta}(-y)=\hat{\theta}(y)$, and it is translation invariant if $\hat{\theta}(y-X \beta)=$ $\hat{\theta}(y)$. Some of the well-known estimators of $\theta$, including ANOVA, ML, and REML estimators (see Sects. 1.3, 1.3.1, and 1.5), are even and translation invariant. In their arguments, however, Kackar and Harville had assumed the existence of the expected value of EBLUP, which is not obvious because, unlike BLUP, EBLUP is not linear in $y$. The existence of the expected value of EBLUP was proved by Jiang (1999b, 2000a). See Sect. 2.7 for details.

Harville (1991) considered the one-way random effects model of Example 1.1 and showed that, in this case, the EBLUP of the mixed effect, $\mu+\alpha_i$, is identical to a parametric empirical Bayes (PEB) estimator. In the meantime, Harville noted some differences between these two approaches, PEB and EBLUP. One of the differences is that much of the work on PEB has been carried out by professional statisticians and has been theoretical in nature. The work has tended to focus on relatively simple models, such as the one-way random effects model, because it is only these models that are tractable from a theoretical standpoint. On the other hand, much of the work on EBLUP has been carried out by practitioners, such as researchers in the animal breeding industry, and has been applied to relatively complex models.

A problem of practical interest is estimation of the MSPE of EBLUP. Such a problem arises, for example, in small area estimation (e.g., Rao and Molina 2015), where the EBLUP is used to estimate the small area means, which can be expressed as mixed effects under a mixed effects model. However, the MSPE of EBLUP is complicated. A naive estimator of the MSPE of EBLUP may be obtained by replacing $\theta$ by $\hat{\theta}$ in the expression of the MSPE of BLUP. However, this is an underestimation. To see this, let $\hat{\eta}=a^{\prime} \hat{\alpha}+b^{\prime} \hat{\beta}$ denote the EBLUP of a mixed effect $\eta=a^{\prime} \alpha+b^{\prime} \beta$, where $\hat{\alpha}$ and $\hat{\beta}$ are the BLUP of $\alpha$, given by (2.35), and BLUE of $\beta$, given by (2.33), respectively, with the variance components $\theta$ replaced by $\hat{\theta}$. Kackar and Harville (1981) showed that, under the normality assumption, one has

$$\operatorname{MSE}(\hat{\eta})=\operatorname{MSE}(\tilde{\eta})+\mathrm{E}(\hat{\eta}-\tilde{\eta})^2$$
where $\tilde{\eta}$ is the BLUP of $\eta$ given by (2.34). It is seen that the MSPE of BLUP is only the first term on the right side of (2.38). In fact, it can be shown that $\operatorname{MSPE}(\tilde{\eta})=$ $g_1(\theta)+g_2(\theta)$, where
\begin{aligned} & g_1(\theta)=a^{\prime}\left(G-G Z^{\prime} V^{-1} Z G\right) a, \ & g_2(\theta)=\left(b-X^{\prime} V^{-1} Z G a\right)^{\prime}\left(X^{\prime} V^{-1} X\right)^{-1}\left(b-X^{\prime} V^{-1} Z G a\right) \end{aligned}

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。