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Ridge-type shrinkage estimators in generalized linear models with an application to prostate cancer data

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Abstract

This paper is concerned with the estimation of the regression coefficients for the generalized linear models in the situation where a multicollinear issue exists and when it is suspected that some of the regression coefficients may be restricted to a linear subspace. Accordingly, as a solution to this issue, we propose a new Stein-type shrinkage ridge estimation approach. We provide the analytic expressions for the asymptotic biases and risks of the proposed estimators and investigate their relative performance with respect to the unrestricted ridge regression estimator. Monte-Carlo simulation studies are conducted to appraise the performance of the underlying estimators in terms of their simulated relative efficiencies. It is clear from both the analytical results and the simulation study that the Stein-type shrinkage ridge estimators dominate the usual ridge regression estimator in the entire parameter space. Finally an empirical application is provided where prostate cancer data is analyzed to show the practical usefulness of the suggested approach. Based on the results from the different parts of this paper, we find that the new method developed would be useful for the practitioners in various research areas such as economics, insurance data and medicine.

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Acknowledgements

We would like to thank the editor and referees for valuable and constructive comments which significantly improved the presentation of paper and led to put many details.

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Authors and Affiliations

  1. Department of Statistics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran

    M. Nooi Asl, H. Bevrani & R. Arabi Belaghi

  2. Department of Economics and Statistics, Jonkoping University, Jönköping, Sweden

    K. Mansson

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  1. M. Nooi Asl
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  2. H. Bevrani
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  3. R. Arabi Belaghi
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  4. K. Mansson
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Correspondence to R. Arabi Belaghi.

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Appendix

Appendix

The technical proofs of the Theorems 3.3 and 3.5 are included in this section. First, we establish the two fundamental Lemmas.

Lemma 1

Let \(\varvec{\mathcal {X}}\) be q-dimensional normal vector distributed as \(\mathcal {N}_{q}(\varvec{\mu }_{x},\varvec{\Sigma }_{q})\), then, for a measurable function of \(\varphi \), we have

$$\begin{aligned} \mathbb {E}\big [\varvec{\mathcal {X}}\varphi (\varvec{\mathcal {X}}^{\prime }\varvec{\mathcal {X}})\big ]&=\varvec{\mu }_{x}\mathbb {E}\big [\varphi \chi _{q+2}^{2}(\Delta ^{2})\big ],\\ \mathbb {E}\big [\varvec{\mathcal {X}}\varvec{\mathcal {X}}^{\prime }\varphi (\varvec{\mathcal {X}}^{\prime }\varvec{\mathcal {X}})\big ]&=\varvec{\Sigma }_{q}\mathbb {E}\big [\varphi \chi _{q+2}^{2}(\Delta ^{2})\big ]+\varvec{\mu }_{x}\varvec{\mu }^{\prime }_{x}\mathbb {E}\big [\varphi \chi _{q+4}^{2}(\Delta ^{2})\big ]. \end{aligned}$$

where \(\Delta ^{2}\) is the non-centrality parameter.

Proof

The proof of this lemma can be found in Judge and Bock (1978). \(\square \)

Lemma 2

Under the assumed regularity conditions and local alternatives \(\varvec{\mathcal {K}}_{(n)}\), as \(n\rightarrow \infty \), we have

$$\begin{aligned}&\Bigg [\begin{array}{cc}\varvec{\eta }_{1n}\\ \varvec{\eta }_{3n}\end{array}\Bigg ]{{\mathop {\longrightarrow }\limits ^{\mathcal {D}}}}\Bigg [\begin{array}{cc}\varvec{\eta }_{1}\\ \varvec{\eta }_{3}\end{array}\Bigg ]\; \sim \mathcal {N}_{2p}\Bigg (\Bigg [\begin{array}{cc}\varvec{0}\\ -\varvec{J}\end{array}\Bigg ],\Bigg [\begin{array}{cc}\varvec{\Sigma }^{-1}&{}\varvec{\Sigma }^{-1}-\varvec{J}_0\\ \varvec{\Sigma }^{-1}-\varvec{J}_0&{}\varvec{\Sigma }^{-1}-\varvec{J}_0\\ \end{array}\Bigg ]\Bigg ),\\&\Bigg [\begin{array}{cc}\varvec{\eta }_{2n}\\ \varvec{\eta }_{3n}\end{array}\Bigg ]{{\mathop {\longrightarrow }\limits ^{\mathcal {D}}}}\Bigg [\begin{array}{cc}\varvec{\eta }_{2}\\ \varvec{\eta }_{3}\end{array}\Bigg ]\; \sim \mathcal {N}_{2p}\Bigg (\Bigg [\begin{array}{cc}-\varvec{J}\\ \varvec{J}\end{array}\Bigg ],\Bigg [\begin{array}{cc}\varvec{\Sigma }^{-1}-\varvec{J}_0&{}\varvec{0}\\ \varvec{0}&{}\varvec{J}_0\\ \end{array}\Bigg ]\Bigg ), \end{aligned}$$

where \(\varvec{\eta }_{1n}=\sqrt{n}(\hat{\varvec{\beta }}^{UE}_{\kappa }-\varvec{\beta })\), \(\varvec{\eta }_{2n}=\sqrt{n}(\hat{\varvec{\beta }}^{RE}_{\kappa }-\varvec{\beta })\), \(\varvec{\eta }_{3n}=\sqrt{n}(\hat{\varvec{\beta }}^{UE}_{\kappa }-\hat{\varvec{\beta }}^{RE}_{\kappa })\), \(\varvec{J}=\varvec{\Sigma }^{-1}\varvec{F}^{\mathsf {T}}(\varvec{F}\varvec{\Sigma }^{-1}\varvec{F}^{\mathsf {T}})^{-1}\varvec{\xi }\) and \(\varvec{J}_0=\varvec{J}\varvec{\xi }^{-1}\varvec{F}\varvec{\Sigma }^{-1}\).

Proof

For detailed proof, see Lisawadi et al. (2016). \(\square \)

1.1 Proof of Theorem 3.3

Here, we provide the proof of the ADB expressions of the proposed estimators. Based on Theorem 3.2, it is clear that

$$\begin{aligned} \text {ADB}\Big (\tilde{\varvec{\beta }}^{UE}_{\kappa }\Big )&=\mathbb {E}\Big [\lim _{n\rightarrow \infty }\sqrt{n}\big (\tilde{\varvec{\beta }}^{UE}_{\kappa }-\varvec{\beta }\big )\Big ]=\mathbb {E}[\varvec{U}^{(1)}]=-\kappa \varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }, \\ \text {ADB}\Big (\tilde{\varvec{\beta }}^{RE}_{\kappa }\Big )&=\mathbb {E}\Big [\lim _{n\rightarrow \infty }\sqrt{n}\big (\tilde{\varvec{\beta }}^{RE}_{\kappa }-\varvec{\beta }\big )\Big ]=\mathbb {E}[\varvec{U}^{(2)}]=-\kappa \varvec{\Sigma }^{-1}_{\kappa }\varvec{\beta }-\varvec{R}(\kappa )\varvec{J}. \end{aligned}$$

Then,

$$\begin{aligned} \text {ADB}\Big (\tilde{\varvec{\beta }}^{JS}_{\kappa }\Big )&=\mathbb {E}\Big [\lim _{n\rightarrow \infty } \sqrt{n}\big (\tilde{\varvec{\beta }}^{JS}_{\kappa }-\varvec{\beta }\big )\Big ]\\&=\mathbb {E}\Big [\lim _{n\rightarrow \infty }\sqrt{n}\Big (\tilde{\varvec{\beta }}^{UE}_{\kappa }-c\varvec{R}_{n}(\kappa )\big (\hat{\varvec{\beta }}^{UE}-\hat{\varvec{\beta }}^{RE}\big )\varvec{\Lambda }_{n}^{-1}-\varvec{\beta }\Big )\Big ]\\&= \mathbb {E}\Big [\lim _{n\rightarrow \infty }\sqrt{n} \big (\tilde{\varvec{\beta }}^{UE}_{\kappa }-\varvec{\beta }\big )\Big ]\\&\quad -c\mathbb {E}\Big [\lim _{n\rightarrow \infty }\sqrt{n}\Big (\varvec{R}_{n}(\kappa )\big (\hat{\varvec{\beta }}^{UE}-\hat{\varvec{\beta }}^{RE}\big )\varvec{\Lambda }_{n}^{-1}\Big )\Big ]\\&= \text {ADB}(\tilde{\varvec{\beta }}^{UE}_{\kappa })-c\varvec{R}(\kappa )\mathbb {E}[\varvec{\eta }^{(3)}\chi ^{-2}_{c+2}(\Delta ^{2})]\\&=-\kappa \varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }-c\varvec{R}(\kappa )\varvec{J}\mathbb {E}[\chi ^{-2}_{c+4}(\Delta ^{2})], \end{aligned}$$

and

$$\begin{aligned} \text {ADB}\Big (\tilde{\varvec{\beta }}^{S+}_{\kappa }\Big )&=\mathbb {E}\Big [\lim _{n\rightarrow \infty } \sqrt{n}\big (\tilde{\varvec{\beta }}^{S+}_{\kappa }-\varvec{\beta }\big )\Big ]\\&=\mathbb {E}\Big [\lim _{n\rightarrow \infty } \sqrt{n}\Big (\tilde{\varvec{\beta }}^{JS}_{\kappa }-\varvec{R}_{n}(\kappa )\big (\hat{\varvec{\beta }}^{UE}-\hat{\varvec{\beta }}^{RE}\big )\big \lbrace 1-c\varvec{\Lambda }_{n}^{-1}\big \rbrace I\big (\varvec{\Lambda }_{n}<c\big )-\varvec{\beta }\Big )\Big ]\\&=\mathbb {E}\Big [\lim _{n\rightarrow \infty } \sqrt{n}\big (\tilde{\varvec{\beta }}^{JS}_{\kappa }-\varvec{\beta }\big )\Big ]\\&\quad -\mathbb {E}\Big [\lim _{n\rightarrow \infty } \sqrt{n}\Big (\varvec{R}_{n}(\kappa )\big (\hat{\varvec{\beta }}^{UE}_{n}-\hat{\varvec{\beta }}^{RE}_{n}\big )\big \lbrace 1-c\varvec{\Lambda }_{n}^{-1}\big \rbrace I\big (\varvec{\Lambda }_{n}<c\big )\Big )\Big ]\\&=\text {ADB}(\tilde{\varvec{\beta }}^{JS}_{\kappa })-\mathbb {E}\Big [\lim _{n\rightarrow \infty } \sqrt{n}\Big (\varvec{R}_{n}(\kappa )\big (\hat{\varvec{\beta }}^{UE}-\hat{\varvec{\beta }}^{RE}\big )I\big (\varvec{\Lambda }_{n}<c\big )\Big )\Big ]\\&\quad +c\mathbb {E}\Big [\lim _{n\rightarrow \infty } \sqrt{n}\Big (\varvec{R}_{n}(\kappa )\big (\hat{\varvec{\beta }}^{UE}-\hat{\varvec{\beta }}^{RE}\big )\varvec{\Lambda }_{n}^{-1} I\big (\varvec{\Lambda }_{n}<c\big )\Big )\Big ]\\&= \text {ADB}(\tilde{\varvec{\beta }}^{JS}_{\kappa })-\varvec{R}(\kappa )\mathbb {E}\Big [\varvec{\eta }^{(3)}I\big (\chi ^{2}_{c+2}(\Delta ^{2})<c\big )\Big ]\\&\quad +c\varvec{R}(\kappa )\mathbb {E}\Big [\varvec{\eta }^{(3)}\chi ^{-2}_{c+2}(\Delta ^{2})I\big (\chi ^{2}_{c+2}(\Delta ^{2})<c\big )\Big ]\\&=-\,\kappa \varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }-\varvec{R}(\kappa )\varvec{J}\mathbb {E}\Big [I\big (\chi ^{2}_{c+4}(\Delta ^{2})<c\big )\Big ]\\&\quad +c\varvec{R}(\kappa )\varvec{J}\mathbb {E}\Big [\chi ^{-2}_{c+4}(\Delta ^{2})I\big (\chi ^{2}_{c+4}(\Delta ^{2})<c\big )\Big ]\\&=-\,\kappa \varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }-\varvec{R}(\kappa )\varvec{J}\Big \lbrace H_{c+4}(c;\Delta ^{2})+c\mathbb {E}\Big [\chi ^{-2}_{c+4}(\Delta ^{2})I\big (\chi ^{2}_{c+4}(\Delta ^{2})>c\big )\Big ]\Big \rbrace . \end{aligned}$$

\(\square \)

1.2 Proof of Theorem 3.5

To prove this theorem, we first derive the asymptotic covariance matrix of the estimators. The asymptotic covariance of an estimator \(\hat{\varvec{\beta }}^{*}\) is defined as follows

$$\begin{aligned} \varvec{V}(\hat{\varvec{\beta }}^{*})=\mathbb {E}\Big [\lim _ {n\rightarrow \infty }\sqrt{n}\big (\hat{\varvec{\beta }}^{*}-\varvec{\beta }\big )\sqrt{n}\big (\hat{\varvec{\beta }}^{*}-\varvec{\beta }\big )^{\mathsf {T}}\Big ]. \end{aligned}$$

First, we derive the covariance matrices of the URRE and RRRE as follows

$$\begin{aligned} \varvec{V}\Big (\tilde{\varvec{\beta }}^{UE}_{\kappa }\Big )&=\mathbb {E}\Big [\lim _{n\rightarrow \infty }\sqrt{n}\big (\tilde{\varvec{\beta }}^{UE}_{\kappa }-\varvec{\beta }\big )\sqrt{n}\big (\tilde{\varvec{\beta }}^{UE}_{\kappa }-\varvec{\beta }\big )^{\mathsf {T}}\Big ]\\&=\mathbb {E}\Big [\lim _{n\rightarrow \infty }\varvec{U}^{(1)}_{n}(\varvec{U}^{(1)}_n)^{\mathsf {T}}\Big ]=\mathbb {E}\big [\varvec{U}^{(1)}(\varvec{U}^{(1)})^{\mathsf {T}}\big ]\\&=\mathbb {V}\big (\varvec{U}^{(1)}\big )+\mathbb {E}[\varvec{U}^{(1)}]\mathbb {E}[(\varvec{U}^{(1)})^{\mathsf {T}}]\\&=\varvec{R}(\kappa )\varvec{\Sigma }^{-1}\varvec{R}(\kappa )+\kappa ^{2}\varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }\varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1},\\ \varvec{V}\Big (\tilde{\varvec{\beta }}^{RE}_{\kappa }\Big )&=\mathbb {E}\Big [\lim _{n\rightarrow \infty }\sqrt{n}\big (\tilde{\varvec{\beta }}^{RE}_{\kappa }-\varvec{\beta }\big )\sqrt{n}\big (\tilde{\varvec{\beta }}^{RE}_{\kappa }-\varvec{\beta }\big )^{\mathsf {T}}\Big ]\\&=\mathbb {E}\Big [\lim _{n\rightarrow \infty }\varvec{U}^{(2)}_{n}(\varvec{U}^{(2)}_n)^{\mathsf {T}}\Big ]= \mathbb {E}\Big [\varvec{U}^{(2)}(\varvec{U}^{(2)})^{\mathsf {T}}\Big ]\\&= \mathbb {V}\big (\varvec{U}^{(2)}\big )+\mathbb {E}[\varvec{U}^{(2)}]\mathbb {E}[(\varvec{U}^{(2)})^{\mathsf {T}}]\\&=\varvec{R}(\kappa )[\varvec{\Sigma }^{-1}-\varvec{J}_0]\varvec{R}(\kappa )+\kappa ^{2}\varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }\varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}+\varvec{R}(\kappa )\varvec{J}\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\\&\quad +\kappa [\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }+\varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\varvec{R}(\kappa )\varvec{J}]. \end{aligned}$$

Next, we obtain \(\varvec{\Sigma }^{*}(\tilde{\varvec{\beta }}^{JS}_{\kappa })\) as follows

$$\begin{aligned} \varvec{V}\Big (\tilde{\varvec{\beta }}^{JS}_{\kappa }\Big )&=\mathbb {E}\Big [\lim _{n\rightarrow \infty }\sqrt{n}\big (\tilde{\varvec{\beta }}^{JS}_{\kappa }-\varvec{\beta }\big )\sqrt{n}\big (\tilde{\varvec{\beta }}^{JS}_{\kappa }-\varvec{\beta }\big )^{\mathsf {T}}\Big ]\\&= \mathbb {E}\Big [\lim _{n\rightarrow \infty }\sqrt{n}\Big \lbrace \tilde{\varvec{\beta }}^{UE}_{\kappa }-c\varvec{R}_{n}(\kappa )\big (\hat{\varvec{\beta }}^{UE}-\hat{\varvec{\beta }}^{RE}\big )\varvec{\Lambda }_{n}^{-1}-\varvec{\beta }\Big \rbrace \\&\quad \times \sqrt{n}\Big \lbrace \tilde{\varvec{\beta }}^{UE}_{\kappa }-c\varvec{R}_{n}(\kappa )\big (\hat{\varvec{\beta }}^{UE}-\hat{\varvec{\beta }}^{RE}\big )\varvec{\Lambda }_{n}^{-1}-\varvec{\beta }\Big \rbrace ^{\mathsf {T}}\Big ]\\&= \mathbb {E}\Big [\lim _{n\rightarrow \infty } \Big \lbrace \varvec{U}^{(1)}_{n}-c\varvec{R}_{n}(\kappa )\varvec{\eta }^{(3)}_{n}\varvec{\Lambda }_{n}^{-1}\Big \rbrace \Big \lbrace \varvec{U}^{(1)}_{n}-c\varvec{R}_{n}(\kappa )\varvec{\eta }^{(3)}_{n}\varvec{\Lambda }_{n}^{-1}\Big \rbrace ^{\mathsf {T}}\Big ]\\&=\mathbb {E}\Big [\lim _{n\rightarrow \infty }\varvec{U}^{(1)}_{n}(\varvec{U}^{(1)}_n)^{\mathsf {T}}\Big ]-c\mathbb {E}\Big [\lim _{n\rightarrow \infty }\varvec{U}^{(1)}_{n}(\varvec{\eta }^{(3)}_n)^{\mathsf {T}}\varvec{\Lambda }_{n}^{-1}\varvec{R}_{n}(\kappa )\Big ]\\&\quad -c\mathbb {E}\Big [\lim _{n\rightarrow \infty }\varvec{R}_{n}(\kappa )\varvec{\eta }^{(3)}_{n}(\varvec{U}^{(1)}_n)^{\mathsf {T}}\varvec{\Lambda }_{n}^{-1}\Big ] +c^{2}\mathbb {E}\Big [\lim _{n\rightarrow \infty }\varvec{R}_{n}(\kappa )\varvec{\eta }^{(3)}_{n}(\varvec{\eta }^{(3)}_n)^{\mathsf {T}}\varvec{\Lambda }_{n}^{-2}\varvec{R}_{n}(\kappa )\Big ]\\&=\underbrace{\mathbb {E}\Big [\varvec{U}^{(1)}(\varvec{U}^{(1)})^{\mathsf {T}}\Big ]} _{E_0} -c\underbrace{\mathbb {E}\Big [\varvec{U}^{(1)}(\varvec{\eta }^{(3)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\Big ]} _{E_1}\varvec{R}(\kappa )\\&\quad -c\varvec{R}(\kappa )\underbrace{\mathbb {E}\Big [\varvec{\eta }^{(3)}(\varvec{U}^{(1)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\Big ]} _{E_2}+c^{2}\varvec{R}(\kappa )\underbrace{\mathbb {E}\Big [\varvec{\eta }^{(3)}(\varvec{\eta }^{(3)})^{\mathsf {T}}\chi ^{-4}_{c+2}(\Delta ^{2})\Big ]} _{E_3}\varvec{R}(\kappa ). \end{aligned}$$

where

$$\begin{aligned} E_{0}&=\mathbb {V}(\varvec{U}^{(1)})+\mathbb {E}[\varvec{U}^{(1)}]\mathbb {E}[(\varvec{U}^{(1)})^{\mathsf {T}}]=\varvec{R}(\kappa )\varvec{\Sigma }^{-1}\varvec{R}(\kappa )+\kappa ^{2}\varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }\varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1},\\ E_{1}&=\varvec{R}(\kappa )\mathbb {E}\Big [\varvec{\eta }^{(1)}(\varvec{\eta }^{(3)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\Big ]+\big (\varvec{R}(\kappa )-\varvec{I}_{p}\big )\varvec{\beta }\mathbb {E}\Big [(\varvec{\eta }^{(3)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\Big ]\\&=\varvec{R}(\kappa )\mathbb {E}\Big [\mathbb {E}\Big (\varvec{\eta }^{(1)}(\varvec{\eta }^{(3)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})|\varvec{\eta }^{(3)}\Big )\Big ]-\kappa \varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }\mathbb {E}\Big [(\varvec{\eta }^{(3)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\Big ]\\&=\varvec{R}(\kappa )\mathbb {E}\Big [\mathbb {E}\big (\varvec{\eta }^{(1)}|\varvec{\eta }^{(3)})(\varvec{\eta }^{(3)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\Big ]-\kappa \varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }\mathbb {E}\Big [(\varvec{\eta }^{(3)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\Big ]\\&=\varvec{R}(\kappa )\mathbb {E}\Big [\big (\varvec{\eta }^{(3)}-\varvec{J}\big )(\varvec{\eta }^{(3)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\Big ]-\kappa \varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }\mathbb {E}\Big [(\varvec{\eta }^{(3)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\Big ]\\&=\varvec{R}(\kappa ) \Big \lbrace \mathbb {E}\big [\varvec{\eta }^{(3)}(\varvec{\eta }^{(3)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\big ]-\varvec{J}\mathbb {E}\big [(\varvec{\eta }^{(3)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\big ]\Big \rbrace \\&\quad -\kappa \varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }\mathbb {E}\Big [(\varvec{\eta }^{(3)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\Big ]. \end{aligned}$$

Using Lemma 1 and the well known identity \(\mathbb {E}[\chi ^{-2}_{k+2}(\Delta ^{2})]-\mathbb {E}[\chi ^{-2}_{k+4}(\Delta ^{2})]=2\mathbb {E}[\chi ^{-4}_{k+4}(\Delta ^{2})]\), \(E_1\) is obtained as

$$\begin{aligned} E_1&=\varvec{R}(\kappa )\varvec{J}_0\mathbb {E}[\chi ^{-2}_{c+4}(\Delta ^{2})]-2\varvec{R}(\kappa )\varvec{J}\varvec{J}^{\mathsf {T}}\mathbb {E}[\chi ^{-4}_{c+4}(\Delta ^{2})]-\kappa \varvec{J}^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }\mathbb {E}[\chi ^{-2}_{c+4}(\Delta ^{2})]. \end{aligned}$$

For \(E_2\), we have

$$\begin{aligned} E_2&=\mathbb {E}\Big [\varvec{\eta }^{(3)}(\varvec{\eta }^{(1)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\Big ]\varvec{R}(\kappa )+\varvec{\beta }^{\mathsf {T}} \big (\varvec{R}(\kappa )-\varvec{I}_{p}\big )\mathbb {E}\Big [\varvec{\eta }^{(3)}\chi ^{-2}_{c+2}(\Delta ^{2})\Big ]\\&=\mathbb {E}\Big [\mathbb {E}\Big (\varvec{\eta }^{(3)}(\varvec{\eta }^{(1)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})| \varvec{\eta }^{(1)}\Big )\Big ]\varvec{R}(\kappa )-\kappa \varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\mathbb {E}\Big [\varvec{\eta }^{(3)}\chi ^{-2}_{c+2}(\Delta ^{2})\Big ]\\&=\mathbb {E}\Big [\mathbb {E}\big (\varvec{\eta }^{(3)}|\varvec{\eta }^{(1)})(\varvec{\eta }^{(1)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\Big ]\varvec{R}(\kappa )-\kappa \varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\varvec{J}\mathbb {E}[\chi ^{-2}_{c+4}(\Delta ^{2})]\\&=\mathbb {E}\Big [\big (\varvec{J}_0\varvec{\Sigma }\varvec{\eta }^{(1)}+\varvec{J}\big )(\varvec{\eta }^{(1)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\Big ]\varvec{R}(\kappa )-\kappa \varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\varvec{J}\mathbb {E}[\chi ^{-2}_{c+4}(\Delta ^{2})]\\&=\mathbb {E} \Big [\varvec{J}_0\varvec{\Sigma }\mathbb {E}\big [\varvec{\eta }^{(1)}(\varvec{\eta }^{(1)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\big ]+\varvec{J}\mathbb {E}\big [(\varvec{\eta }^{(1)})^{\mathsf {T}}\chi ^{-2}_{c+2}(\Delta ^{2})\big ]\Big ]\varvec{R}(\kappa )\\&\quad -\kappa \varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\varvec{J}\mathbb {E}[\chi ^{-2}_{c+4}(\Delta ^{2})]\\&=\varvec{J}_0\mathbb {E}[\chi ^{-2}_{c+4}(\Delta ^{2})]\varvec{R}(\kappa )-\kappa \varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\varvec{J}\mathbb {E}[\chi ^{-2}_{c+4}(\Delta ^{2})]. \end{aligned}$$

and

$$\begin{aligned} E_3&=\varvec{J}_0\mathbb {E}[\chi ^{-4}_{c+4}(\Delta ^{2})]+\varvec{J}\varvec{J}^{\mathsf {T}}\mathbb {E}[\chi ^{-4}_{c+6}(\Delta ^{2})]. \end{aligned}$$

Then, \(\varvec{V}(\tilde{\varvec{\beta }}^{JS}_{\kappa })\) can be written in form of

$$\begin{aligned} \varvec{V}(\tilde{\varvec{\beta }}^{JS}_{\kappa })&=\varvec{R}(\kappa )\Big [\varvec{\Sigma }^{-1}-c\varvec{J}_0\Big \lbrace 2\mathbb {E}[\chi ^{-2}_{c+4}(\Delta ^{2})]-c\mathbb {E}[\chi ^{-4}_{c+4}(\Delta ^{2})]\Big \rbrace \Big ]\varvec{R}(\kappa )\\&\quad +c\kappa \big [\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }+\varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\varvec{R}(\kappa )\varvec{J}\big ]\mathbb {E}[\chi ^{-2}_{c+4}(\Delta ^{2})]\\&\quad +c(c+2)\varvec{R}(\kappa )\varvec{J}\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\mathbb {E}[\chi ^{-4}_{c+6}(\Delta ^{2})]+\kappa ^{2}\varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }\varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}. \end{aligned}$$

Finally, \(\varvec{V}\Big (\tilde{\varvec{\beta }}^{\text {S+}}_{\kappa }\Big )\) can be obtained as follows

$$\begin{aligned} \varvec{V}\Big (\tilde{\varvec{\beta }}^{S+}_{\kappa }\Big )&=\mathbb {E}\Big [\lim _{n\rightarrow \infty } \sqrt{n}\big (\tilde{\varvec{\beta }}^{S+}_{\kappa }-\varvec{\beta }\big )\sqrt{n}\big (\tilde{\varvec{\beta }}^{S+}_{\kappa }-\varvec{\beta }\big )^{\mathsf {T}}\Big ]\\&=\mathbb {E}\Bigg [\lim _{n\rightarrow \infty }\Big \lbrace \sqrt{n}\big (\tilde{\varvec{\beta }}^{JS}_{\kappa }-\varvec{\beta }\big )-\varvec{R}_{n}(\kappa ) \big [1-c\varvec{\Lambda }_{n}^{-1}\big ]\varvec{\eta }^{(3)}_{n}I(\varvec{\Lambda }_{n}<c)\Big \rbrace \\&\quad \times \Big \lbrace \sqrt{n}\big (\tilde{\varvec{\beta }}^{JS}_{\kappa }-\varvec{\beta }\big )-\varvec{R}_{n}(\kappa )\big [1-c\varvec{\Lambda }_{n}^{-1}\big ]\varvec{\eta }^{(3)}_{n}I(\varvec{\Lambda }_{n}<c)\Big \rbrace ^{\mathsf {T}}\Bigg ]\\&=\varvec{V}(\tilde{\varvec{\beta }}^{JS}_{\kappa })-\mathbb {E}\Big [\lim _{n\rightarrow \infty }\sqrt{n}\big (\tilde{\varvec{\beta }}^{JS}_{\kappa }-\varvec{\beta }\big )\big (\varvec{\eta }^{(3)}_{n}\big )^{\mathsf {T}}\big [1-c\varvec{\Lambda }_{n}^{-1}\big ]I(\varvec{\Lambda }_{n}<c)\varvec{R}_{n}(\kappa )\Big ]\\&\quad -\mathbb {E}\Big [\lim _{n\rightarrow \infty }\varvec{R}_{n}(\kappa )\varvec{\eta }^{(3)}_{n}\Big \lbrace \sqrt{n}\big (\tilde{\varvec{\beta }}^{JS}_{\kappa }-\varvec{\beta }\big )\Big \rbrace ^{\mathsf {T}}\big [1-c\varvec{\Lambda }_{n}^{-1}\big ]I(\varvec{\Lambda }_{n}<c)\Big ]\\&\quad +\mathbb {E}\Big [\lim _{n\rightarrow \infty }\varvec{R}_{n}(\kappa )\varvec{\eta }^{(3)}_{n}(\varvec{\eta }^{(3)}_{n})^{\mathsf {T}}\big [1-c\varvec{\Lambda }_{n}^{-1}\big ]^{2}I(\varvec{\Lambda }_{n}<c)\varvec{R}_{n}(\kappa )\Big ]\\&=\varvec{V}(\tilde{\varvec{\beta }}^{JS}_{\kappa })-\mathbb {E}\Big [\lim _{n\rightarrow \infty }\sqrt{n}\big (\tilde{\varvec{\beta }}^{UE}_{\kappa }-\varvec{\beta }\big )(\varvec{\eta }^{(3)}_{n})^{\mathsf {T}}\big [1-c\varvec{\Lambda }_{n}^{-1}\big ]I(\varvec{\Lambda }_{n}<c)\varvec{R}_{n}(\kappa )\Big ]\\&\quad -\mathbb {E}\Big [\lim _{n\rightarrow \infty }\varvec{R}_{n}(\kappa )\varvec{\eta }^{(3)}_{n}\Big \lbrace \sqrt{n}\big (\tilde{\varvec{\beta }}^{UE}_{\kappa }-\varvec{\beta }\big )\Big \rbrace ^{\mathsf {T}}\big [1-c\varvec{\Lambda }_{n}^{-1}\big ]I(\varvec{\Lambda }_{n}<c)\Big ]\\&\quad +2c\mathbb {E}\Big [\lim _{n\rightarrow \infty }\varvec{R}_{n}(\kappa )\varvec{\eta }^{(3)}_{n}(\varvec{\eta }^{(3)}_{n})^{\mathsf {T}}\varvec{\varvec{\Lambda }}_{n}^{-1}\big [1-c\varvec{\Lambda }_{n}^{-1}\big ]I(\varvec{\Lambda }_{n}<c)\varvec{R}_{n}(\kappa )\Big ]\\&\quad +\mathbb {E}\Big [\lim _{n\rightarrow \infty }\varvec{R}_{n}(\kappa )\varvec{\eta }^{(3)}_{n}(\varvec{\eta }^{(3)}_{n})^{\mathsf {T}}\big [1-c\varvec{\Lambda }_{n}^{-1}\big ]^{2}I(\varvec{\Lambda }_{n}<c)\varvec{R}_{n}(\kappa )\Big ]\\&= \varvec{V}(\tilde{\varvec{\beta }}^{JS}_{\kappa })-\mathbb {E}\Big [\lim _{n\rightarrow \infty }\varvec{U}^{(1)}_{n}(\varvec{\eta }^{(3)}_{n})^{\mathsf {T}}\big [1-c\varvec{\Lambda }_{n}^{-1}\big ]I(\varvec{\Lambda }_{n}<c)\varvec{R}_{n}(\kappa )\Big ]\\&\quad -\mathbb {E}\Big [\lim _{n\rightarrow \infty }\varvec{R}_{n}(\kappa )\varvec{\eta }^{(3)}_{n}(\varvec{U}^{(1)}_{n})^{\mathsf {T}}\big [1-c\varvec{\Lambda }_{n}^{-1}\big ]I(\varvec{\Lambda }_{n}<c)\Big ]\\&\quad +2\mathbb {E}\Big [\lim _{n\rightarrow \infty }\varvec{R}_{n}(\kappa )\varvec{\eta }^{(3)}_{n}(\varvec{\eta }^{(3)}_{n})^{\mathsf {T}}\big [1-c\varvec{\Lambda }_{n}^{-1}\big ]I(\varvec{\Lambda }_{n}<c)\varvec{R}_{n}(\kappa )\Big ]\\&\quad -\mathbb {E}\Big [\lim _{n\rightarrow \infty }\varvec{R}_{n}(\kappa )\varvec{\eta }^{(3)}_{n}(\varvec{\eta }^{(3)}_{n})^{\mathsf {T}}\big [1-c\varvec{\Lambda }_{n}^{-1}\big ]^{2}I(\varvec{\Lambda }_{n}<c)\varvec{R}_{n}(\kappa )\Big ]\\&=\varvec{V}(\tilde{\varvec{\beta }}^{JS}_{\kappa })-\underbrace{\mathbb {E}\Big [\varvec{U}^{(1)}(\varvec{\eta }^{(3)})^{\mathsf {T}}\big [1-c\chi ^{-2}_{c+2}(\Delta ^{2})\big ]I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]} _{E4}\varvec{R}(\kappa )\\&\quad -\varvec{R}(\kappa )\underbrace{\mathbb {E}\Big [\varvec{\eta }^{(3)}(\varvec{U}^{(1)})^{\mathsf {T}}\big [1-c\chi ^{-2}_{c+2}(\Delta ^{2})\big ]I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]} _{E_5}\\&\quad +2\varvec{R}(\kappa )\mathbb {E}\Big [\varvec{\eta }^{(3)}(\varvec{\eta }^{(3)})^{\mathsf {T}}\big [1-c\chi ^{-2}_{c+2}(\Delta ^{2})\big ]I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\varvec{R}(\kappa )\\&\quad -\varvec{R}(\kappa )\mathbb {E}\Big [\varvec{\eta }^{(3)}(\varvec{\eta }^{(3)})^{\mathsf {T}}\big [1-c\chi ^{-2}_{c+2}(\Delta ^{2})\big ]^{2}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\varvec{R}(\kappa ), \end{aligned}$$

where

$$\begin{aligned} E_4&=\varvec{R}(\kappa )\mathbb {E}\Big [\mathbb {E}\Big [\varvec{\eta }^{(1)}(\varvec{\eta }^{(3)})^{\mathsf {T}}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)|\varvec{\eta }^{(3)}\Big ]\Big ]\\&\quad -\kappa \varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }\mathbb {E}\Big [(\varvec{\eta }^{(3)})^{\mathsf {T}}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\\&=\varvec{R}(\kappa )\mathbb {E}\Big [\mathbb {E}\big [\varvec{\eta }^{(1)}|\varvec{\eta }^{(3)}\big ](\varvec{\eta }^{(3)})^{\mathsf {T}}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\\&\quad -\kappa \varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }\mathbb {E}\Big [(\varvec{\eta }^{(3)})^{\mathsf {T}}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\\&=\varvec{R}(\kappa )\mathbb {E}\Big [\varvec{\eta }^{(3)}(\varvec{\eta }^{(3)})^{\mathsf {T}}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\\&\quad -\varvec{R}(\kappa )\varvec{J}\mathbb {E}\Big [(\varvec{\eta }^{(3)})^{\mathsf {T}}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\\&\quad -\kappa \varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }\mathbb {E}\Big [(\varvec{\eta }^{(3)})^{\mathsf {T}}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ],\\ E_5&=\mathbb {E}\Big [\mathbb {E}\Big [\varvec{\eta }^{(3)}(\varvec{\eta }^{(1)})^{\mathsf {T}}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)|\varvec{\eta }^{(1)}\Big ]\Big ]\varvec{R}(\kappa )\\&\quad -\kappa \varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\mathbb {E}\Big [\varvec{\eta }^{(3)}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\\&=\mathbb {E}\Big [\mathbb {E}\big [\varvec{\eta }^{(3)}|\varvec{\eta }^{(1)}\big ](\varvec{\eta }^{(1)})^{\mathsf {T}}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\varvec{R}(\kappa )\\&\quad -\kappa \varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\mathbb {E}\Big [\varvec{\eta }^{(3)}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\\&=\varvec{J}_0\varvec{\Sigma }\mathbb {E}\Big [\varvec{\eta }^{(1)}(\varvec{\eta }^{(1)})^{\mathsf {T}}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\varvec{R}(\kappa )\\&\quad +\varvec{J}\mathbb {E}\Big [(\varvec{\eta }^{(1)})^{\mathsf {T}}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\varvec{R}(\kappa )\\&\quad -\kappa \varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\mathbb {E}\Big [\varvec{\eta }^{(3)}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]. \end{aligned}$$

By using Lemma 1, we obtain

$$\begin{aligned} E_5&=\varvec{J}\mathbb {E}\big [\{1-c\chi ^{-2}_{c+4}(\Delta ^{2})\}I(\chi ^{2}_{c+4}(\Delta ^{2})<c)\big ]\varvec{R}(\kappa )\\&\quad -\kappa \varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\varvec{J}\mathbb {E}\big [\{1-c\chi ^{-2}_{c+4}(\Delta ^{2})\}I(\chi ^{2}_{c+4}(\Delta ^{2})<c)\big ]. \end{aligned}$$

Thus,

$$\begin{aligned} \varvec{V}\Big (\tilde{\varvec{\beta }}^{S+}_{\kappa }\Big )&=\varvec{V}(\tilde{\varvec{\beta }}^{JS}_{\kappa })-\varvec{R}(\kappa )\mathbb {E}\Big [\varvec{\eta }^{(3)}(\varvec{\eta }^{(3)})^{\mathsf {T}}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\varvec{R}(\kappa )\\&\quad +\varvec{R}(\kappa )\varvec{J}\mathbb {E}\Big [(\varvec{\eta }^{(3)})^{\mathsf {T}}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\varvec{R}(\kappa )\\&\quad +\kappa \varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }\mathbb {E}\Big [(\varvec{\eta }^{(3)})^{\mathsf {T}}\{1-c\chi ^{-2}_{c+2}(\Delta ^{2})\}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\varvec{R}(\kappa )\\&\quad -\varvec{R}(\kappa )\varvec{J}_0\mathbb {E}\big [\{1-c\chi ^{-2}_{c+4}(\Delta ^{2})\}I(\chi ^{2}_{c+4}(\Delta ^{2})<c)\big ]\varvec{R}(\kappa )\\&\quad +\kappa \varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\varvec{R}(\kappa )\varvec{J}\mathbb {E}\big [\{1-c\chi ^{-2}_{c+4}(\Delta ^{2})\}I(\chi ^{2}_{c+4}(\Delta ^{2})<c)\big ]\\&\quad +2\varvec{R}(\kappa )\mathbb {E}\Big [\varvec{\eta }^{(3)}(\varvec{\eta }^{(3)})^{\mathsf {T}}\big [1-c\chi ^{-2}_{c+2}(\Delta ^{2})\big ]I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\varvec{R}(\kappa )\\&\quad -\varvec{R}(\kappa )\mathbb {E}\Big [\varvec{\eta }^{(3)}(\varvec{\eta }^{(3)})^{\mathsf {T}}\big [1-c\chi ^{-2}_{c+2}(\Delta ^{2})\big ]^{2}I(\chi ^{2}_{c+2}(\Delta ^{2})<c)\Big ]\varvec{R}(\kappa ). \end{aligned}$$

Finally,

$$\begin{aligned} \varvec{V}\Big (\tilde{\varvec{\beta }}^{S+}_{\kappa }\Big )&=\varvec{V}\Big (\tilde{\varvec{\beta }}^{JS}_{\kappa }\Big )-\varvec{R}(\kappa )\varvec{J}_0\varvec{R}(\kappa )\mathbb {E}\big [\{1-c\chi ^{-2}_{c+4}(\Delta ^{2})\}I(\chi ^{2}_{c+4}(\Delta ^{2})<c)\big ]\\&\quad +\varvec{R}(\kappa )\varvec{J}\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\Bigg \lbrace \mathbb {E}\big [\{1-c\chi ^{-2}_{c+6}(\Delta ^{2})\}I(\chi ^{2}_{c+6}(\Delta ^{2})<c)\big ]\\&\quad +\mathbb {E}\big [\{1-c\chi ^{-2}_{c+4}(\Delta ^{2})\}I(\chi ^{2}_{c+4}(\Delta ^{2})<c)\big ]\\&\quad -\mathbb {E}\big [\{1-c\chi ^{-2}_{c+6}(\Delta ^{2})\}^{2}I(\chi ^{2}_{c+6}(\Delta ^{2})<c)\big ]\Bigg \rbrace \\&\quad +\kappa [\varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }^{-1}_{\kappa }\varvec{R}(\kappa )\varvec{J}+\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\varvec{\Sigma }^{-1}_{\kappa }\varvec{\beta }] \mathbb {E}\big [\{1-c\chi ^{-2}_{c+4}(\Delta ^{2})\}I(\chi ^{2}_{c+4}(\Delta ^{2})<c)\big ]. \end{aligned}$$

Now, we provide the proof of the ADR expressions of the proposed estimators.

$$\begin{aligned} \mathcal {R}\Big (\tilde{\varvec{\beta }}^{UE}_{\kappa };\varvec{Q}\Big )&=trace\big (\varvec{Q}\varvec{V}(\tilde{\varvec{\beta }}^{UE}_{\kappa })\big )=trace\big (\varvec{Q}\varvec{R}(\kappa )\varvec{\Sigma }^{-1}\varvec{R}(\kappa )\big )+\kappa ^{2}\varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }^{-1}_{\kappa }\varvec{Q}\varvec{\Sigma }^{-1}_{\kappa }\varvec{\beta }. \end{aligned}$$

Then,

$$\begin{aligned} \mathcal {R}\Big (\tilde{\varvec{\beta }}^{RE}_{\kappa };\varvec{Q}\Big )&=trace\big (\varvec{Q}\varvec{V}(\tilde{\varvec{\beta }}^{RE}_{\kappa })\big )=\mathcal {R}(\tilde{\varvec{\beta }}^{UE}_{\kappa };\varvec{Q})-trace\big (\varvec{Q}\varvec{R}(\kappa )\varvec{J}_0\varvec{R}(\kappa )\big )\\&\quad +trace\big (\varvec{Q}\varvec{R}(\kappa )\varvec{J}\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\big )\!+\! \kappa [\varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }^{-1}_{\kappa }\varvec{Q}\varvec{R}(\kappa )\varvec{J}\!+\!\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\varvec{Q}\varvec{\Sigma }^{-1}_{\kappa }\varvec{\beta }]\\&=\mathcal {R}(\tilde{\varvec{\beta }}^{UE}_{\kappa };\varvec{Q})-trace\big (\varvec{Q}\varvec{R}(\kappa )\varvec{J}_0\varvec{R}(\kappa )\big )\\&\quad +\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\varvec{Q}\varvec{R}(\kappa )\varvec{J}+\kappa [\varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }^{-1}_{\kappa }\varvec{Q}\varvec{R}(\kappa )\varvec{J}+\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\varvec{Q}\varvec{\Sigma }^{-1}_{\kappa }\varvec{\beta }]. \end{aligned}$$

Also

$$\begin{aligned} \mathcal {R}\Big (\tilde{\varvec{\beta }}^{JS}_{\kappa };\varvec{Q}\Big )&=trace\big (\varvec{Q}\varvec{V}(\tilde{\varvec{\beta }}^{JS}_{\kappa })\big )\\&=\mathcal {R}(\tilde{\varvec{\beta }}^{UE}_{\kappa };\varvec{Q})- trace\big (\varvec{Q}\varvec{R}(\kappa )\varvec{J}_0\varvec{R}(\kappa )\big )\Big \lbrace 2c\mathbb {E}[\chi ^{-2}_{c+4}(\Delta ^{2})]-c^{2}\mathbb {E}[\chi ^{-4}_{c+4}(\Delta ^{2})]\Big \rbrace \\&\quad +c\kappa \big [\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\varvec{Q}\varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }+\varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\varvec{Q}\varvec{R}(\kappa )\varvec{J}\big ]\mathbb {E}[\chi ^{-2}_{c+4}(\Delta ^{2})]\\&\quad +c(c+2)trace\big (\varvec{Q}\varvec{R}(\kappa )\varvec{J}\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\big )\mathbb {E}[\chi ^{-4}_{c+6}(\Delta ^{2})]\\&=\mathcal {R}(\tilde{\varvec{\beta }}^{UE}_{\kappa };\varvec{Q})-trace\big (\varvec{Q}\varvec{R}(\kappa )\varvec{J}_0\varvec{R}(\kappa )\big )\Big \lbrace 2c\mathbb {E}[\chi ^{-2}_{c+4}(\Delta ^{2})]-c^{2}\mathbb {E}[\chi ^{-4}_{c+4}(\Delta ^{2})]\Big \rbrace \\&\quad +c\kappa \big [\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\varvec{Q}\varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }+\varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\varvec{Q}\varvec{R}(\kappa )\varvec{J}\big ]\mathbb {E}[\chi ^{-2}_{c+4}(\Delta ^{2})]\\&\quad +c(c+2)\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\varvec{Q}\varvec{R}(\kappa )\varvec{J}\mathbb {E}[\chi ^{-4}_{c+6}(\Delta ^{2})]. \end{aligned}$$

Finally,

$$\begin{aligned}&\mathcal {R}\Big (\tilde{\varvec{\beta }}^{S+}_{\kappa };\varvec{Q}\Big )=trace\big (\varvec{Q}\varvec{V}(\tilde{\varvec{\beta }}^{S+}_{\kappa })\big )=trace\big (\varvec{Q}{V}(\tilde{\varvec{\beta }}^{JS}_{\kappa })\big )\\&\quad -trace\big (\varvec{Q}\varvec{R}(\kappa )\varvec{J}_0\varvec{R}(\kappa )\big )\mathbb {E}\big [\{1-c\chi ^{-2}_{c+4}(\Delta ^{2})\}I(\chi ^{2}_{c+4}(\Delta ^{2})<c)\big ]\\&\quad +trace\big (\varvec{Q}\varvec{R}(\kappa )\varvec{J}\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\big )\Bigg \lbrace \mathbb {E}\big [\{1-c\chi ^{-2}_{c+6}(\Delta ^{2})\}I(\chi ^{2}_{c+6}(\Delta ^{2})<c)\big ]\\&\quad +\mathbb {E}\big [\{1-c\chi ^{-2}_{c+4}(\Delta ^{2})\}I(\chi ^{2}_{c+4}(\Delta ^{2})<c)\big ]\\&\quad -\mathbb {E}\big [\{1-c\chi ^{-2}_{c+6}(\Delta ^{2})\}^{2}I(\chi ^{2}_{c+6}(\Delta ^{2})<c)\big ]\Bigg \rbrace \\&\quad +\kappa \big [\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\varvec{Q}\varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }+\varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\varvec{Q}\varvec{R}(\kappa )\varvec{J}\big ]\mathbb {E}\big [\{1-c\chi ^{-2}_{c+4}(\Delta ^{2})\}I(\chi ^{2}_{c+4}(\Delta ^{2})<c)\big ]\\&=\mathcal {R}(\tilde{\varvec{\beta }}^{JS}_{\kappa };\varvec{Q})-trace\big (\varvec{Q}\varvec{R}(\kappa )\varvec{J}_0\varvec{R}(\kappa )\big )\mathbb {E}\big [\{1-c\chi ^{-2}_{c+4}(\Delta ^{2})\}I(\chi ^{2}_{c+4}(\Delta ^{2})<c)\big ]\\&\quad +\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\varvec{Q}\varvec{R}(\kappa )\varvec{J}\Bigg \lbrace \mathbb {E}\big [\{1-c\chi ^{-2}_{c+6}(\Delta ^{2})\}I(\chi ^{2}_{c+6}(\Delta ^{2})<c)\big ]\\&\quad +\mathbb {E}\big [\{1-c\chi ^{-2}_{c+4}(\Delta ^{2})\}I(\chi ^{2}_{c+4}(\Delta ^{2})<c)\big ]\\&\quad -\mathbb {E}\big [\{1-c\chi ^{-2}_{c+6}(\Delta ^{2})\}^{2}I(\chi ^{2}_{c+6}(\Delta ^{2})<c)\big ]\Bigg \rbrace \\&\quad +\kappa \big [\varvec{J}^{\mathsf {T}}\varvec{R}(\kappa )\varvec{Q}\varvec{\Sigma }_{\kappa }^{-1}\varvec{\beta }+\varvec{\beta }^{\mathsf {T}}\varvec{\Sigma }_{\kappa }^{-1}\varvec{Q}\varvec{R}(\kappa )\varvec{J}\big ]\mathbb {E}\big [\{1-c\chi ^{-2}_{c+4}(\Delta ^{2})\}I(\chi ^{2}_{c+4}(\Delta ^{2})<c)\big ]. \end{aligned}$$

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Nooi Asl, M., Bevrani, H., Arabi Belaghi, R. et al. Ridge-type shrinkage estimators in generalized linear models with an application to prostate cancer data. Stat Papers 62, 1043–1085 (2021). https://doi.org/10.1007/s00362-019-01123-w

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