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

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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。

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