### 统计代写|似然估计作业代写Probability and Estimation代考|Estimation of Ridge Parameter

statistics-lab™ 为您的留学生涯保驾护航 在代写似然估计Probability and Estimation方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写似然估计Probability and Estimation代写方面经验极为丰富，各种代写似然估计Probability and Estimation相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|似然估计作业代写Probability and Estimation代考|Estimation of Ridge Parameter

We can observe from Eq. (1.18) that the RRE heavily depends on the ridge parameter $k$. Many authors at different times worked in this area of research and developed and proposed different estimators for $k$. They considered various models such as linear regression, Poisson regression, and logistic regression models. To mention a few, Hoerl and Kennard (1970), Hoerl et al. (1975), McDonald and Galarneau (1975), Lawless and Wang (1976), Dempster et al. $(1977)$, Gibbons (1981), Kibria (2003), Khalaf and Shukur (2005), Alkhamisi and Shukur (2008), Muniz and Kibria (2009), Gruber et al. (2010), Muniz et al. (2012), Mansson et al. (2010), Hefnawy and Farag (2013), Aslam (2014), and Arashi and Valizadeh (2015), and Kibria and Banik (2016), among others.

## 统计代写|似然估计作业代写Probability and Estimation代考| Preliminary Test and Stein-Type Ridge Estimators

In previous sections, we discussed the notion of RRE and how it shrinks the elements of the ordinary LSE. Sometimes, it is needed to shrink the LSE to a subspace defined by $\boldsymbol{H} \boldsymbol{\beta}=\boldsymbol{h}$, where $\boldsymbol{H}$ is a $q \times p$ known matrix of full row rank $q(q \leq p)$ and $h$ is a $q$ vector of known constants. It is also termed as constraint or restriction. Such a configuration of the subspace is frequently used in the design of experiments, known as contrasts. Therefore, sometimes shrinking is for two purposes. We refer to this as double shrinking.

In general, unlike the Bayesian paradigm, correctness of the prior information $\boldsymbol{H} \boldsymbol{\beta}=\boldsymbol{h}$ can be tested on the basis of samples through testing $\mathcal{H}{\circ}: \boldsymbol{H} \boldsymbol{\beta}=\boldsymbol{h}$ vs. a set of alternatives. Following Fisher’s recipe, we use the non-sample information $\boldsymbol{H} \boldsymbol{\beta}=\boldsymbol{h}$; if based on the given sample, we accept $\mathcal{H}{\mathrm{o}}$. In situations where this prior information is correct, an efficient estimator is the one which satisfies this restriction, called the restricted estimator.

To derive the restricted estimator under a multicollinear situation, satisfying the condition $\boldsymbol{H} \boldsymbol{\beta}=\boldsymbol{h}$, one solves the following convex optimization problem,
$$\min {\boldsymbol{\beta}}\left{\mathrm{PS}{\lambda}(\boldsymbol{\beta})\right}, \quad \mathrm{PS}{\lambda}(\boldsymbol{\beta})=(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})^{\top}(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})+k|\boldsymbol{\beta}|^{2}+\lambda^{\top}(\boldsymbol{H} \boldsymbol{\beta}-\boldsymbol{h}),$$ where $\lambda=\left(\lambda{1}, \ldots, \lambda_{q}\right)^{\top}$ is the vector of Lagrangian multipliers. Grob (2003) proposed the restricted RRE, under a multicollinear situation, by correcting the restricted RRE of Sarkar (1992).

In our case, we consider prior information with the form $\boldsymbol{\beta}=\mathbf{0}$, which is a test used for checking goodness of fit. Here, the restricted RRE is simply given by $\hat{\boldsymbol{\beta}}{n}(k)=\mathbf{0}$, where $\mathbf{0}$ is the restricted estimator of $\boldsymbol{\beta}$. Therefore, one uses $\hat{\boldsymbol{\beta}}{n}^{\mathrm{RR}}(k)$ if $\mathcal{H}{\mathrm{o}}: \boldsymbol{\beta}=\mathbf{0}$ rejects and $\hat{\boldsymbol{\beta}}{n}^{\mathrm{RR}(\mathrm{R})}(k)$ if $\mathcal{H}{\mathrm{o}}: \boldsymbol{\beta}=\mathbf{0}$ accepts. Combining the information existing in both estimators, one may follow an approach by Bancroft (1964), to propose the preliminary test RRE given by \begin{aligned} \hat{\boldsymbol{\beta}}{n}^{\mathrm{RR}(\mathrm{PT})}(k, \alpha) &=\left{\begin{array}{cl} \hat{\boldsymbol{\beta}}{n}^{\mathrm{RR}}(k) ; & \mathcal{H}{\mathrm{o}} \text { is rejected } \ \hat{\boldsymbol{\beta}}{n}^{\mathrm{RR}(\mathrm{R})}(k) ; & \mathcal{H}{\mathrm{o}} \text { is accepted } \end{array}\right.\ &=\hat{\boldsymbol{\beta}}{n}^{\mathrm{RR}}(k) I\left(\mathcal{H}{\mathrm{a}} \text { is rejected }\right)+\hat{\boldsymbol{\beta}}{n}^{\mathrm{RR}(\mathrm{R})}(k) I\left(\mathcal{H}{\mathrm{o}} \text { is accepted }\right) \ &=\hat{\boldsymbol{\beta}}{n}^{\mathrm{RR}}(k) I\left(\mathcal{L}{n}>F_{p, m}(\alpha)\right)+\hat{\boldsymbol{\beta}}{n}^{\mathrm{RR}(\mathrm{R})}(k) I\left(\mathcal{L}{n} \leq F_{p, m}(\alpha)\right) \ &=\hat{\boldsymbol{\beta}}{n}^{\mathrm{RR}}(k)-\hat{\boldsymbol{\beta}}{n}^{\mathrm{RR}}(k) I\left(\mathcal{L}{n} \leq F{p, m}(\alpha)\right), \end{aligned}
where $I(A)$ is the indicator function of the set $A$ and $\mathcal{L}{n}$ is the test statistic for testing $\mathcal{H}{\mathrm{o}}: \boldsymbol{\beta}=\mathbf{0}$, and $F_{q, m}(\alpha)$ is the upper $\alpha$-level critical value from the $F$-distribution with $(q, m)$ degrees of freedom (D.F.) See Judge and Bock (1978) and Saleh (2006) for the test statistic and details.

## 统计代写|似然估计作业代写Probability and Estimation代考| Notes and References

The first paper on ridge analysis was by Hoerl (1962); however, the first paper on multicollinearity appeared five years later, roughly speaking, by Farrar and Glauber (1967). Marquardt and Snee (1975) reviewed the theory of ridge regression and its relation to generalized inverse regression. Their study includes several illustrative examples about ridge regression. For the geometry of multicollinearity, see Akdeniz and Ozturk (1981). We also suggest that Gunst (1983) and and Sakallioglu and Akdeniz (1998) not be missed. Gruber (1998) in his monograph motivates the need for using ridge regression and allocated a large portion to the analysis of ridge regression and its generalizations. For historical survey up to 1998 , we refer to Gruber (1998).

Beginning from 2000 , a comprehensive study in ridge regression is the work of Ozturk and Akdeniz (2000), where the authors provide some solutions for ill-posed inverse problems. Wan (2002) incorporated measure of goodness of fit in evaluating the RRE and proposed a feasible generalized RRE. Kibria

(2003) gave a comprehensive analysis about the estimation of ridge parameter $k$ for the linear regression model. For application of ridge regression in agriculture, see Jamal and Rind (2007). Maronna (2011) proposed an RRE based on repeated M-estimation in robust regression. Saleh et al. (2014) extensively studied the performance of preliminary test and Stein-type ridge estimators in the multivariate-t regression model. Huang et al. (2016) defined a weighted VIF for collinearity diagnostic in generalized linear models.

Arashi et al. (2017) studied the performance of several ridge parameter estimators in a restricted ridge regression model with stochastic constraints. Asar et al. (2017) defined a restricted RRE in the logistic regression model and derived its statistical properties. Roozbeh and Arashi (2016a) developed a new ridge estimator in partial linear models. Roozbeh and Arashi (2016b) used difference methodology to study the performance of an RRE in a partial linear model. Arashi and Valizadeh (2015) compared several estimators for estimating the biasing parameter in the study of partial linear models in the presence of multicollinearity. Roozbeh and Arashi (2013) proposed a feasible RRE in partial linear models and studied its properties in details. Roozbeh et al. (2012) developed RREs in seemingly partial linear models. Recently, Chandrasekhar et al. (2016) proposed the concept of partial ridge regression, which involves selectively adjusting the ridge constants associated with highly collinear variables to control instability in the variances of coefficient estimates. Norouzirad and Arashi (2017) developed shrinkage ridge estimators in the context of robust regression. Fallah et al. (2017) studied the asymptotic performance of a general form of shrinkage ridge estimator. Recently, Norouzirad et al. (2017) proposed improved robust ridge $M$-estimators and studied their asymptotic behavior.

## 统计代写|似然估计作业代写Probability and Estimation代考| Preliminary Test and Stein-Type Ridge Estimators

\min {\boldsymbol{\beta}}\left{\mathrm{PS}{\lambda}(\boldsymbol{\beta})\right}, \quad \mathrm{PS}{\lambda}(\boldsymbol{\ beta})=(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})^{\top}(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})+k| \boldsymbol{\beta}|^{2}+\lambda^{\top}(\boldsymbol{H} \boldsymbol{\beta}-\boldsymbol{h}),\min {\boldsymbol{\beta}}\left{\mathrm{PS}{\lambda}(\boldsymbol{\beta})\right}, \quad \mathrm{PS}{\lambda}(\boldsymbol{\ beta})=(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})^{\top}(\boldsymbol{Y}-\boldsymbol{X} \boldsymbol{\beta})+k| \boldsymbol{\beta}|^{2}+\lambda^{\top}(\boldsymbol{H} \boldsymbol{\beta}-\boldsymbol{h}),在哪里λ=(λ1,…,λq)⊤是拉格朗日乘数的向量。Grob (2003) 通过修正 Sarkar (1992) 的受限 RRE，提出了在多重共线性情况下的受限 RRE。

## 统计代写|似然估计作业代写Probability and Estimation代考| Notes and References

（2003）对岭参数的估计进行了综合分析ķ对于线性回归模型。关于岭回归在农业中的应用，请参见 Jamal 和 Rind（2007 年）。Maronna (2011) 在稳健回归中提出了一种基于重复 M 估计的 RRE。萨利赫等人。（2014 年）广泛研究了多元 t 回归模型中初步测试和 Stein 型岭估计器的性能。黄等人。(2016) 为广义线性模型中的共线性诊断定义了加权 VIF。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。