Appendix
We present the following two lemmas below, which will enable us to derive the results of Theorems 1 and 3 in this paper
Lemma 1
If \(k/\sqrt{n}\rightarrow \lambda _{0}\ge 0\) and \(\varvec{ {\tilde{Q}}}\) is non-singular, then
$$\begin{aligned} \sqrt{n}\left( \widehat{\varvec{\beta }}^{\mathrm{FM}}-{\varvec{\beta }}\right) \overset{d}{\rightarrow }\mathcal {N}\left( -\lambda _{0} \varvec{{\tilde{Q}}}^{-1}{\varvec{\beta }}\text {, }\,\sigma ^{2} \varvec{{\tilde{Q}}}^{-1}\right) , \end{aligned}$$
where “\(\overset{d}{\rightarrow }\)” denotes convergence in distribution.
Proof
Let define \(V_n({{\mathbf {u}}})\) as follows:
$$\begin{aligned} \sum _{ i=1 }^{ n }{ \left[ { \left( {\tilde{\varepsilon }}_{ i }-{{\mathbf {u}}}'{\tilde{{{\mathbf {x}}}}}_{i}/\sqrt{ n } \right) }^{ 2 }-{\tilde{\varepsilon }}_{ i }^2 \right] +k\sum _{ j=1 }^{ p }{ \left[ { \left| \beta _j +u_j/\sqrt{ n } \right| }^{ 2 }-{ \left| \beta _j \right| }^{ 2 } \right] } }, \end{aligned}$$
where \({{\mathbf {u}}}=(u_1,\dots ,u_p)'\). Following Knight and Fu (2000), it can be shown that
$$\begin{aligned} \sum _{ i=1 }^{ n }{ \left[ { \left( {\tilde{\varepsilon }}_{ i }-{{\mathbf {u}}}'{\tilde{{{\mathbf {x}}}}}_{i}/\sqrt{ n } \right) }^{ 2 }-{\tilde{\varepsilon }}_{ i }^2 \right] } \overset{d}{\rightarrow } -2{{\mathbf {u}}}'{{\mathbf {D}}}+{{\mathbf {u}}}'\varvec{{\tilde{Q}}}{{\mathbf {u}}}, \end{aligned}$$
where \(\mathbf{D}\sim \mathcal {N}(\mathbf{0},\sigma ^2{{\mathbf {I}}}_p)\), with finite-dimensional convergence holding trivially. Hence,
$$\begin{aligned} k\sum _{ j=1 }^{ p }{ \left[ { \left| \beta _j +u_j/\sqrt{ n } \right| }^{ 2 }-{ \left| \beta _j \right| }^{ 2 } \right] } \overset{d}{\rightarrow } \lambda _{0}\sum _{ j=1 }^{ p }u_j \text{ sgn }(\beta _j)|\beta _j|. \end{aligned}$$
Hence, \(V_n({{\mathbf {u}}})\overset{d}{\rightarrow }V({{\mathbf {u}}})\). Because \(V_n\) is convex and V has a unique minimum, by following Geyer (1996), it yields
$$\begin{aligned} {\mathrm{arg\, min}}(V_n)= \sqrt{n}\left( \widehat{\varvec{\beta }}^{\mathrm{FM}}-{\varvec{\beta }}\right) \overset{d}{\rightarrow } {\mathrm{arg\, min}}(V). \end{aligned}$$
Hence,
$$\begin{aligned} \sqrt{n}\left( \widehat{\varvec{\beta }}^{\mathrm{FM}}-{\varvec{\beta }}\right) \overset{d}{\rightarrow } \varvec{{\tilde{Q}}}^{-1} \left( {{\mathbf {D}}}- \lambda _{0}{\varvec{\beta }}\right) {\sim }\mathcal {N}\left( -\lambda _{0} \varvec{{\tilde{Q}}}^{-1}{\varvec{\beta }}\text {, }\,\sigma ^{2} \varvec{{\tilde{Q}}}^{-1}\right) . \end{aligned}$$
\(\square \)
Lemma 2
Let \({{\mathbf {X}}}\) be \(q-\)dimensional normal vector distributed as \( \mathcal {N}\left( \varvec{\mu }_{x},\varvec{{\varSigma } } _{q}\right) \), then, for a measurable function of \(\varphi ,\) we have
$$\begin{aligned} \text {E}\,\left[ {{\mathbf {X}}}\varphi \left( {{\mathbf {X}}}'{{\mathbf {X}}}\right) \right]&=\varvec{\mu }_{x}\text {E}\,\left[ \varphi \chi _{q+2}^{2}\left( {\varDelta } \right) \right] \\ \text {E}\,\left[ {{\mathbf {X}}}{{\mathbf {X}}}'\varphi \left( {{\mathbf {X}}}' {{\mathbf {X}}}\right) \right]&=\varvec{{\varSigma } }_{q}\text {E}\,\left[ \varphi \chi _{q+2}^{2}\left( {\varDelta } \right) \right] +\varvec{\mu }_{x}\varvec{\mu }_{x}'\text {E}\,\left[ \varphi \chi _{q+4}^{2}\left( {\varDelta } \right) \right] \end{aligned}$$
where \(\chi _{v}^{2}\left( {\varDelta } \right) \) is a non-central chi-square distribution with v degrees of freedom and non-centrality parameter \({\varDelta }\).
Proof
It can be found in Judge and Bock (1978) \(\square \)
We further consider the following proposition for proving theorems.
Proposition 1
Under local alternative \(\left\{ K_{n}\right\} \) as \(n\rightarrow \infty \), we have
$$\begin{aligned}&\left( \begin{array}{c} \vartheta _{1} \\ \vartheta _{3} \end{array} \right) \sim \mathcal {N}\left[ \left( \begin{array}{c} -\varvec{\eta }_{11.2} \\ \varvec{\delta } \end{array} \right) ,\left( \begin{array}{cc} \sigma ^{2}\varvec{{\tilde{Q}}}_{11.2}^{-1} &{} \varvec{{\varPhi } }_{*} \\ \varvec{{\varPhi } }_{*} &{} \varvec{{\varPhi } }_{*} \end{array} \right) \right] ,\\&\left( \begin{array}{c} \vartheta _{3} \\ \vartheta _{2} \end{array} \right) \sim \mathcal {N}\left[ \left( \begin{array}{c} \varvec{\delta } \\ -\varvec{\xi } \end{array} \right) ,\left( \begin{array}{cc} \varvec{{\varPhi } }_{*} &{} \varvec{0} \\ \varvec{0} &{} \sigma ^{2}\varvec{{\tilde{Q}}}_{11}^{-1} \end{array} \right) \right] , \end{aligned}$$
where \(\vartheta _{1} =\sqrt{n}\left( \widehat{\varvec{\beta }}_{1}^{\mathrm{FM}}- {\varvec{\beta }}_{1}\right) \), \(\vartheta _{2} = \sqrt{n}\left( \widehat{\varvec{\beta }}_{1}^{\mathrm{SM}}- {\varvec{\beta }}_{1}\right) \) and \(\vartheta _{3}=\vartheta _{1}-\vartheta _{2}\).
Proof
Under the light of Lemmas 1 and 2, it can easily be obtained
$$\begin{aligned} \vartheta _{1} \overset{d}{\rightarrow }\mathcal {N}\left( -\varvec{\eta }_{11.2},\sigma ^{2}\varvec{{\tilde{Q}}} _{11.2}^{-1}\right) . \end{aligned}$$
Define \({{\mathbf {y}}}^{*}=\tilde{{{\mathbf {y}}}}-{\tilde{{{\mathbf {X}}}}}_{2}\widehat{\varvec{\beta }}_{2}^{\mathrm{FM}}\), and
$$\begin{aligned} \widehat{\varvec{\beta }}_{1}^{\mathrm{FM}}= & {} \underset{\varvec{\varvec{\beta }_{1}}}{\arg \min }\left\{ \left\| {{\mathbf {y}}}^{*}-{\tilde{{{\mathbf {X}}}}}_{1} {\varvec{\beta }}_{1}\right\| +k\left\| \varvec{\beta }_{1}\right\| ^{2}\right\} \nonumber \\= & {} \left( {\tilde{{{\mathbf {X}}}}}_{1}'{\tilde{{{\mathbf {X}}}}}_{1}+k {{\mathbf {I}}}_{p_{1}}\right) ^{-1}{\tilde{{{\mathbf {X}}}}}_{1}'{{\mathbf {y}}}^{*} \nonumber \\= & {} \left( {\tilde{{{\mathbf {X}}}}}_{1}'{\tilde{{{\mathbf {X}}}}}_{1}+k {{\mathbf {I}}}_{p_{1}}\right) ^{-1}{\tilde{{{\mathbf {X}}}}}_{1}'\tilde{{{\mathbf {y}}}} -\left( {\tilde{{{\mathbf {X}}}}}_{1}'{\tilde{{{\mathbf {X}}}}}_{1}+k{{\mathbf {I}}}_{{p}_{1}}\right) ^{-1}{\tilde{{{\mathbf {X}}}}}_{1}'{\tilde{{{\mathbf {X}}}}}_{2} \widehat{\varvec{\beta }}_{2}^{\mathrm{FM}} \nonumber \\= & {} \widehat{\varvec{\beta }}_{1}^{\mathrm{SM}}-\left( {\tilde{{{\mathbf {X}}}}}_{1}' {\tilde{{{\mathbf {X}}}}}_{1}+k{{\mathbf {I}}}_{p_{1}}\right) ^{-1} {\tilde{{{\mathbf {X}}}}}_{1}'{\tilde{{{\mathbf {X}}}}}_{2}\widehat{\varvec{\beta }}_{2}^{\mathrm{FM}}. \end{aligned}$$
(11)
By using Eq. (11),
$$\begin{aligned} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{SM}} -{\varvec{\beta }}_{1}\right) \right\}= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{FM}} +\varvec{{\tilde{Q}}}_{11}^{-1}\varvec{{\tilde{Q}}}_{12} \varvec{\widehat{\beta }}_{2}^{\mathrm{FM}} -\varvec{\beta }_{1}\right) \right\} \\= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{\beta }_{1}\right) \right\} \\&+\,\text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ {\tilde{Q}}}_{11}^{-1}\varvec{{\tilde{Q}}}_{12}\varvec{\widehat{\beta }}_{2}^{\mathrm{FM}} \right) \right\} \end{aligned}$$
by Lemma 2,
$$\begin{aligned}= & {} -\varvec{\eta }_{11.2}+\varvec{{\tilde{Q}}}_{11}^{-1}\varvec{{\tilde{Q}}} _{12}\varvec{\omega } \\= & {} -\left( \varvec{\eta }_{11.2}-\varvec{\delta }\right) \\= & {} -\varvec{\xi }. \end{aligned}$$
Hence, \(\vartheta _{2} \overset{d}{\rightarrow }\mathcal {N}\left( -\varvec{\varvec{\xi }},\sigma ^{2}\varvec{{\tilde{Q}}}_{11}^{-1}\right) . \)
Using the Eq. (11), we can obtain \(\varvec{{\varPhi } }_{*}\) as follows:
$$\begin{aligned} \varvec{{\varPhi } }_{*}= & {} \text {Cov}\,\left( \varvec{\widehat{\beta }}_{1}^{\mathrm{FM}}- \widehat{\varvec{\beta }}_{1}^{\mathrm{SM}}\right) \\= & {} \text {E}\,\left[ \left( \widehat{\varvec{\beta }}_{1}^{\mathrm{FM}} -\widehat{\varvec{\beta }}_{1}^{\mathrm{SM}}\right) \left( \widehat{\varvec{\beta }}_{1}^{\mathrm{FM}}-\widehat{\varvec{\beta }}_{1}^{\mathrm{SM}}\right) '\right] \\= & {} \text {E}\,\left[ \left( \varvec{{\tilde{Q}}}_{11}^{-1}\varvec{{\tilde{Q}}}_{12}\varvec{ \widehat{\beta }}_{2}^{\mathrm{FM}}\right) \left( \varvec{{\tilde{Q}}}_{11}^{-1}\varvec{{\tilde{Q}}} _{12}\varvec{\widehat{\beta }}_{2}^{\mathrm{FM}}\right) '\right] \\= & {} \varvec{{\tilde{Q}}}_{11}^{-1}\varvec{{\tilde{Q}}}_{12}\text {E}\,\left[ \varvec{\widehat{\beta } }_{2}^{\mathrm{FM}}\left( \varvec{\widehat{\beta }}_{2}^{\mathrm{FM}}\right) '\right] \varvec{{\tilde{Q}}}_{21}\varvec{{\tilde{Q}}}_{11}^{-1} \\= & {} \sigma ^{2}\varvec{{\tilde{Q}}}_{11}^{-1}\varvec{{\tilde{Q}}}_{12}\varvec{{\tilde{Q}}} _{22.1}^{-1}\varvec{{\tilde{Q}}}_{21}\varvec{{\tilde{Q}}}_{11}^{-1} \text {.}\, \end{aligned}$$
We also know that
$$\begin{aligned} \varvec{{\varPhi } }_{*}= & {} \sigma ^{2}\varvec{{\tilde{Q}}}_{11}^{-1}\varvec{{\tilde{Q}}}_{12}\varvec{{\tilde{Q}}} _{22.1}^{-1}\varvec{{\tilde{Q}}}_{21}\varvec{{\tilde{Q}}}_{11}^{-1} = \sigma ^{2} \left( \varvec{{\tilde{Q}}}_{11.2}^{-1}- \varvec{{\tilde{Q}}}_{11}^{-1}\right) . \end{aligned}$$
Hence, it is obtained \(\vartheta _{3} \overset{d}{\rightarrow }\mathcal {N}\left( \varvec{\delta },\varvec{{\varPhi } }_{*}\right) .\)\(\square \)
Proof (Theorem 1)
\(\text {ADB}\,\left( \varvec{\widehat{\beta }}_{1}^{\mathrm{FM}} \right) \) and \(\text {ADB}\,\left( \varvec{\widehat{\beta }}_{1}^{\mathrm{SM}} \right) \) are directly obtained from Proposition 1. Also, the ADBs of PT, S and PS are obtained as follows:
$$\begin{aligned} \text {ADB}\,\left( \varvec{\widehat{\beta }}_{1}^{\mathrm{PT}} \right)= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{PT}} -\varvec{\beta }_{1}\right) \right\} \\= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{FM}} -\left( \varvec{\widehat{\beta }}_{1}^{\mathrm{FM}} - \varvec{\widehat{\beta }}_{1}^{\mathrm{SM}} \right) \text {I}\,\left( \mathcal {T}_{n}\le c_{n,\alpha }\right) -\varvec{\beta }_{1}\right) \right\} \\= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{\beta }_{1}\right) \right\} \\&-\text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{\widehat{\beta }}_{1}^{\mathrm{SM}} \right) \text {I}\,\left( \mathcal {T}_{n}\le c_{n,\alpha }\right) \right) \right\} \\= & {} -\varvec{\eta }_{11.2}-\varvec{\delta }\text {H}\,_{p_{2}+2}\left( \chi _{p_{2},\alpha }^{2};{\varDelta } \right) \text {.}\,\\ \text {ADB}\,\left( \varvec{\widehat{\beta }}_{1}^{\mathrm{S}} \right)= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{S}} -\varvec{\beta } _{1}\right) \right\} \\= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{FM}} -\left( \varvec{\widehat{\beta }}_{1}^{\mathrm{FM}} - \varvec{\widehat{\beta }}_{1}^{\mathrm{SM}} \right) \left( p_{2}-2\right) \mathcal {T}_{n}^{-1}-\varvec{\beta }_{1}\right) \right\} \\= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{\beta }_{1}\right) \right\} \\&-\text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{\widehat{\beta }}_{1}^{\mathrm{SM}} \right) \left( p_{2}-2\right) \mathcal {T}_{n}^{-1}\right) \right\} \\= & {} -\varvec{\eta }_{11.2}-\left( p_{2}-2\right) \varvec{\delta } \text {E}\,\left( \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \right) \text {.}\, \\ \text {ADB}\,\left( \varvec{\widehat{\beta }}_{1}^{\mathrm{PS}} \right)= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{PS}} -\varvec{\beta }_{1}\right) \right\} \\= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{SM}} +\left( \varvec{\widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{ \widehat{\beta }}_{1}^{\mathrm{SM}} \right) \right. \right. \\&\times \left. \left. \left( 1-\left( p_{2}-2\right) T _{n}^{-1}\right) \text {I}\,\left( \mathcal {T}_{n}>p_{2}-2\right) -\varvec{\beta }_{1}\right) \right\} \\= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left[ \varvec{ \widehat{\beta }}_{1}^{\mathrm{SM}} +\left( \varvec{\widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{ \widehat{\beta }}_{1}^{\mathrm{SM}} \right) \left( 1-\text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right) \right. \right. \\&\left. \left. -\left( \varvec{\widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{\widehat{\beta }}_{1}^{\mathrm{SM}} \right) \left( p_{2}-2\right) \mathcal {T}_{n}^{-1}\text {I}\,\left( \mathcal {T}_{n}>p_{2}-2\right) -\varvec{\beta }_{1}\right] \right\} \\= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{\beta }_{1}\right) \right\} \\&-\text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{\widehat{\beta }}_{1}^{\mathrm{SM}} \right) \text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right\} \\&-\text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{\widehat{\beta }}_{1}^{\mathrm{SM}} \right) \left( p_{2}-2\right) \mathcal {T}_{n}^{-1}\text {I}\,\left( \mathcal {T}_{n}>p_{2}-2\right) \right\} \\= & {} -\varvec{\eta }_{11.2}-\varvec{\delta }\text {H}\,_{p_{2}+2}\left( p_{2}-2;\left( {\varDelta } \right) \right) \\&-\,\varvec{\delta }\left( p_{2}-2\right) \text {E}\,\left\{ \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \text {I}\,\left( \chi _{p_{2+2}}^{2}\left( {\varDelta } \right) >p_{2}-2\right) \right\} \text {.}\, \end{aligned}$$
\(\square \)
The asymptotic covariance of an estimator \({\varvec{\beta }}_{1}^{*}\) is defined as follows:
$$\begin{aligned} \text {Cov}\, \left( {\varvec{\beta }}_{1}^{*} \right)= & {} \text {E}\,\left\{ \underset{ n\rightarrow \infty }{\lim }n\left( {\varvec{\beta }}_{1}^{*} -{\varvec{\beta }}_{1}\right) \left( {\varvec{\beta }}_{1}^{*} -\varvec{\beta }_{1}\right) '\right\} \text {.}\, \end{aligned}$$
Proof (Theorem 2)
Firstly, the asymptotic covariance of \(\varvec{\widehat{\beta }}_{1}^{\mathrm{FM}}\) is given by
$$\begin{aligned} \text {Cov}\, \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{FM}} \right)= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{\widehat{ \beta }}_{1}^{\mathrm{FM}} -\varvec{\beta }_{1}\right) \sqrt{n}\left( \varvec{\widehat{ \beta }}_{1}^{\mathrm{FM}} -\varvec{\beta }_{1}\right) '\right\} \\= & {} \text {E}\,\left( \vartheta _{1}\vartheta _{1}'\right) \\= & {} \text {Cov}\,\left( \vartheta _{1}\vartheta _{1}'\right) +\text {E}\,\left( \vartheta _{1}\right) \text {E}\,\left( \vartheta _{1}'\right) \\= & {} \sigma ^{2}\varvec{{\tilde{Q}}}_{11.2}^{-1}+\varvec{\eta }_{11.2}\varvec{ \eta }_{11.2}' \text {.}\, \end{aligned}$$
The asymptotic covariance of \(\varvec{\widehat{\beta }}_{1}^{\mathrm{SM}}\) is given by
$$\begin{aligned} \text {Cov}\, \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{SM}} \right)= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{\widehat{\beta }} _{1}^{\mathrm{SM}} -\varvec{\beta }_{1}\right) \sqrt{n}\left( \varvec{\widehat{\beta }} _{1}^{\mathrm{SM}} -\varvec{\beta }_{1}\right) '\right\} \\= & {} \text {E}\,\left( \vartheta _{2}\vartheta _{2}'\right) \\= & {} \text {Cov}\,\left( \vartheta _{2}\vartheta _{2}'\right) +\text {E}\,\left( \vartheta _{2}\right) \text {E}\,\left( \vartheta _{2}'\right) \\= & {} \sigma ^{2}\varvec{{\tilde{Q}}}_{11}^{-1}+\varvec{\xi \xi }', \end{aligned}$$
The asymptotic covariance of \(\varvec{\widehat{\beta }}_{1}^{\mathrm{PT}}\) is given by
$$\begin{aligned} \text {Cov}\, \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{PT}} \right)= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{PT}} -\varvec{\beta }_{1}\right) \sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{PT}} -\varvec{\beta }_{1}\right) '\right\} \\= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }n\left[ \left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{\beta }_{1}\right) -\left( \varvec{\widehat{ \beta }}_{1}^{\mathrm{FM}} -\varvec{\widehat{\beta }}_{1}^{\mathrm{SM}} \right) \text {I}\,\left( \mathcal {T}_{n}\le c_{n,\alpha }\right) \right] \right. \\&\left. \left[ \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{FM}} -{\varvec{\beta }}_{1}\right) -\left( \varvec{\widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{ \widehat{\beta }}_{1}^{\mathrm{SM}} \right) \text {I}\,\left( \mathcal {T}_{n}\le c_{n,\alpha }\right) \right] '\right\} \\= & {} \text {E}\,\left\{ \left[ \vartheta _{1}-\vartheta _{3}\text {I}\,\left( \mathcal {T}_{n}\le c_{n,\alpha }\right) \right] \left[ \vartheta _{1}-\vartheta _{3}\text {I}\,\left( \mathcal {T}_{n}\le c_{n,\alpha }\right) \right] '\right\} \\= & {} \text {E}\,\left\{ \vartheta _{1}\vartheta _{1}'-2\vartheta _{3}\vartheta _{1}'\text {I}\,\left( \mathcal {T}_{n}\le c_{n,\alpha }\right) +\vartheta _{3}\vartheta _{3}'\text {I}\,\left( \mathcal {T}_{n}\le c_{n,\alpha }\right) \right\} \text {.}\, \end{aligned}$$
Now, by using Lemma 2 and the formula for a conditional mean of a bivariate normal, we have
$$\begin{aligned} \text {E}\,\left\{ \vartheta _{3}\vartheta _{1}'\text {I}\,\left( \mathcal {T}_{n}\le c_{n,\alpha }\right) \right\}= & {} \text {E}\,\left\{ \text {E}\,\left( \vartheta _{3}\vartheta _{1}'\text {I}\,\left( \mathcal {T} _{n}\le c_{n,\alpha }\right) |\vartheta _{3}\right) \right\} \\= & {} \text {E}\,\left\{ \vartheta _{3}\text {E}\,\left( \vartheta _{1}'\text {I}\,\left( \mathcal {T} _{n}\le c_{n,\alpha }\right) |\vartheta _{3}\right) \right\} \\= & {} \text {E}\,\left\{ \vartheta _{3}\left[ -\eta _{11.2}+\left( \vartheta _{3}- \varvec{\delta }\right) \right] '\text {I}\,\left( \mathcal {T}_{n}\le c_{n,\alpha }\right) \right\} \\= & {} -\text {E}\,\left\{ \vartheta _{3}\varvec{\eta }_{11.2}'\text {I}\,\left( \mathcal {T}_{n}\le c_{n,\alpha }\right) \right\} + \\&\text {E}\,\left\{ \vartheta _{3}\left( \vartheta _{3}-\varvec{\delta }\right) '\text {I}\,\left( \mathcal {T}_{n}\le c_{n,\alpha }\right) \right\} \\= & {} -\varvec{\eta }_{11.2}'\text {E}\,\left\{ \vartheta _{3}\text {I}\,\left( \mathcal {T}_{n}\le c_{n,\alpha }\right) \right\} \\&+\text {E}\,\left\{ \vartheta _{3}\vartheta _{3}'\text {I}\,\left( \mathcal {T}_{n}\le c_{n,\alpha }\right) \right\} \\&-\,\text {E}\,\left\{ \vartheta _{3}\varvec{\delta }'\text {I}\,\left( \mathcal {T} _{n}\le c_{n,\alpha }\right) \right\} \\= & {} -\varvec{\eta }_{11.2}'\varvec{\delta }\text {H}\,_{p_{2}+2}\left( \chi _{p_{2},\alpha }^{2};{\varDelta } \right) +\left\{ \text {Cov}\,(\vartheta _{3}\vartheta _{3}')\text {H}\,_{p_{2}+2}\left( \chi _{p_{2},\alpha }^{2};{\varDelta } \right) \right. \\&\left. +\,\text {E}\,\left( \vartheta _{3}\right) \text {E}\,\left( \vartheta _{3}'\right) \text {H}\,_{p_{2}+4}\left( \chi _{p_{2},\alpha }^{2};{\varDelta } \right) - \varvec{\delta \delta }'\text {H}\,_{p_{2}+2}\left( \chi _{p_{2},\alpha }^{2};{\varDelta } \right) \right\} \\= & {} -\varvec{\eta }_{11.2}'\varvec{\delta }\text {H}\,_{p_{2}+2}\left( \chi _{p_{2},\alpha }^{2};{\varDelta } \right) +\varvec{{\varPhi } }_{*}\text {H}\,_{p_{2}+2}\left( \chi _{p_{2},\alpha }^{2};{\varDelta } \right) \\&+\,\varvec{\delta \delta }'\text {H}\,_{p_{2}+4}\left( \chi _{p_{2},\alpha }^{2};{\varDelta } \right) -\varvec{\delta \delta }'\text {H}\,_{p_{2}+2}\left( \chi _{p_{2},\alpha }^{2};{\varDelta } \right) , \end{aligned}$$
then,
$$\begin{aligned} \text {Cov}\, \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{PT}} \right)= & {} \varvec{\eta }_{11.2}\varvec{\eta }_{11.2}'+2\varvec{\eta }_{11.2}'\varvec{\delta }\text {H}\,_{p_{2}+2}\left( \chi _{p_{2},\alpha }^{2};{\varDelta } \right) \\&\sigma ^{2}\varvec{{\tilde{Q}}}_{11.2}^{-1}-\varvec{{\varPhi } }_{*}\text {H}\,_{p_{2}+2}\left( \chi _{p_{2},\alpha }^{2};\left( {\varDelta } \right) \right) - \varvec{\delta \delta }'\text {H}\,_{p_{2}+4}\left( \chi _{p_{2},\alpha }^{2}; {\varDelta } \right) \\&+\,2\varvec{\delta \delta }'\text {H}\,_{p_{2}+2}\left( \chi _{p_{2},\alpha }^{2}; {\varDelta } \right) \\= & {} \sigma ^{2}\varvec{{\tilde{Q}}}_{11.2}^{-1}+\varvec{\eta }_{11.2}\varvec{ \eta }_{11.2}'+2\varvec{\eta }_{11.2}'\varvec{\delta } \text {H}\,_{p_{2}+2}\left( \chi _{p_{2},\alpha }^{2};{\varDelta } \right) \\&+\,\sigma ^{2}\left( \varvec{{\tilde{Q}}}_{11.2}^{-1}- \varvec{{\tilde{Q}}}_{11}^{-1}\right) \text {H}\,_{p_{2}+2}\left( \chi _{p_{2},\alpha }^{2}; {\varDelta } \right) \\&+\,\varvec{\delta \delta }'\left[ 2\text {H}\,_{p_{2}+2}\left( \chi _{p_{2},\alpha }^{2};{\varDelta } \right) -\text {H}\,_{p_{2}+4}\left( \chi _{p_{2},\alpha }^{2};{\varDelta } \right) \right] . \end{aligned}$$
The asymptotic covariance of \(\widehat{\varvec{\beta }}_{1}^{\mathrm{S}}\) is given by
$$\begin{aligned} \text {Cov}\, \left( \widehat{\varvec{\beta }}_{1}^{\mathrm{S}}\right)= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{S}}-{\varvec{\beta }}_{1}\right) \sqrt{n}\left( \widehat{\varvec{\beta }}_{1}^{\mathrm{S}}-{\varvec{\beta }}_{1}\right) '\right\} \\= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }n\left[ \left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{FM}}-{\varvec{\beta }}_{1}\right) -\left( \widehat{\varvec{\beta }}_{1}^{\mathrm{FM}}-\varvec{\widehat{\beta }} _{1}^{\mathrm{SM}}\right) \left( p_{2}-2\right) \mathcal {T}_{n}^{-1}\right] \right. \\&\left. \left[ \left( \widehat{\varvec{\beta }}_{1}^{\mathrm{FM}}-\varvec{ \beta }_{1}\right) -\left( \varvec{\widehat{\beta }}_{1}^{\mathrm{FM}}-\varvec{ \widehat{\beta }}_{1}^{\mathrm{SM}}\right) \left( p_{2}-2\right) \mathcal {T}_{n}^{-1} \right] '\right\} \\= & {} \text {E}\,\left\{ \vartheta _{1}\vartheta _{1}'-2\left( p_{2}-2\right) \vartheta _{3}\vartheta _{1}'\mathcal {T}_{n}^{-1}+\left( p_{2}-2\right) ^{2}\vartheta _{3}\vartheta _{3}'\mathcal {T} _{n}^{-2}\right\} \text {.}\, \end{aligned}$$
Note that, by using Lemma 2 and the formula for a conditional mean of a bivariate normal, we have
$$\begin{aligned} \text {E}\,\left\{ \vartheta _{3}\vartheta _{1}'\mathcal {T}_{n}^{-1}\right\}= & {} \text {E}\,\left\{ \text {E}\,\left( \vartheta _{3}\vartheta _{1}'\mathcal {T} _{n}^{-1}|\vartheta _{3}\right) \right\} \\= & {} \text {E}\,\left\{ \vartheta _{3}\text {E}\,\left( \vartheta _{1}'\mathcal {T} _{n}^{-1}|\vartheta _{3}\right) \right\} \\= & {} \text {E}\,\left\{ \vartheta _{3}\left[ -\varvec{\eta }_{11.2}+\left( \vartheta _{3}-\varvec{\delta }\right) \right] '\mathcal {T} _{n}^{-1}\right\} \\= & {} -\text {E}\,\left\{ \vartheta _{3}\varvec{\eta }_{11.2}'\mathcal {T} _{n}^{-1}\right\} +\text {E}\,\left\{ \vartheta _{3}\left( \vartheta _{3}-\varvec{ \delta }\right) '\mathcal {T}_{n}^{-1}\right\} \\= & {} -\varvec{\eta }_{11.2}'\text {E}\,\left\{ \vartheta _{3}T _{n}^{-1}\right\} +\text {E}\,\left\{ \vartheta _{3}\vartheta _{3}'\mathcal {T} _{n}^{-1}\right\} \\&-\,\text {E}\,\left\{ \vartheta _{3}\varvec{\delta }'\mathcal {T} _{n}^{-1}\right\} \\= & {} -\varvec{\eta }_{11.2}'\varvec{\delta }\text {E}\,\left( \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \right) +\left\{ \text {Cov}\,(\vartheta _{3}\vartheta _{3}')\text {E}\,\left( \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \right) \right. \\&\left. +\,\text {E}\,\left( \vartheta _{3}\right) \text {E}\,\left( \vartheta _{3}'\right) \text {E}\,\left( \chi _{p_{2}+4}^{-2}\left( {\varDelta } \right) \right) - \varvec{\delta \delta }'\text {H}\,_{p_{2}+2}\left( \chi _{p_{2},\alpha }^{2};{\varDelta } \right) \right\} \\= & {} -\varvec{\eta }_{11.2}'\varvec{\delta }\text {E}\,\left( \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \right) +\varvec{{\varPhi } }_{*}\text {E}\,\left( \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \right) \\&+\varvec{\delta \delta }'\text {E}\,\left( \chi _{p_{2}+4}^{-2}\left( {\varDelta } \right) \right) -\varvec{\delta \delta }'\text {E}\,\left( \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \right) ,\\ \text {Cov}\, \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{S}} \right)= & {} \sigma ^{2}\varvec{{\tilde{Q}}}_{11.2}^{-1}+\varvec{\eta }_{11.2}\varvec{ \eta }_{11.2}'+2\left( p_{2}-2\right) \varvec{\eta }_{11.2}' \varvec{\delta }\text {E}\,\left( \chi _{p_{2+2},\alpha }^{-2}\left( {\varDelta } \right) \right) \\&-\left( p_{2}-2\right) \varvec{{\varPhi } }_{*}\left\{ 2\text {E}\,\left( \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \right) -\left( p_{2}-2\right) \text {E}\,\left( \chi _{p_{2+2}}^{-4}\left( {\varDelta } \right) \right) \right\} \\&+\left( p_{2}-2\right) \varvec{\delta \delta }'\left\{ -2\text {E}\,\left( \chi _{p_{2+4}}^{-2}\left( {\varDelta } \right) \right) +2\text {E}\,\left( \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \right) \right. \\&\left. +\left( p_{2}-2\right) \text {E}\,\left( \chi _{p_{2+4}}^{-4}\left( {\varDelta } \right) \right) \right\} \text {.}\, \end{aligned}$$
Finally, the asymptotic covariance matrix of positive shrinkage ridge regression estimator is derived as follows:
$$\begin{aligned} \text {Cov}\, \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{PS}} \right)= & {} \text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }n\left( \varvec{\widehat{ \beta }}_{1}^{\mathrm{PS}} -\varvec{\beta }_{1}\right) \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{PS}} -\varvec{\beta }_{1}\right) '\right\} \\= & {} \text {Cov}\, \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{S}} \right) -2\text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left[ \left( \varvec{ \widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{\widehat{\beta }}_{1}^{\mathrm{SM}} \right) \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{S}} -\varvec{\beta }_{1}\right) '\right. \right. \\&\times \left. \left. \left\{ 1-\left( p_{2}-2\right) T _{n}^{-1}\right\} \text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right] \right\} \\&+\,\text {E}\,\left\{ \underset{n\rightarrow \infty }{\lim }\sqrt{n}\left[ \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{\widehat{\beta }}_{1}^{\mathrm{SM}} \right) \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{FM}} -\varvec{\widehat{\beta } }_{1}^{\mathrm{SM}} \right) '\right. \right. \\&\times \left. \left. \left\{ 1-\left( p_{2}-2\right) T _{n}^{-1}\right\} ^{2}\text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right] \right\} \\= & {} \text {Cov}\, \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{S}} \right) -2\text {E}\,\left\{ \vartheta _{3}\vartheta _{1}'\left\{ 1-\left( p_{2}-2\right) \mathcal {T}_{n}^{-1}\right\} \text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right\} \\&+\,2\text {E}\,\left\{ \vartheta _{3}\vartheta _{3}'\left( p_{2}-2\right) \mathcal {T}_{n}^{-1}\text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right\} \\&-\,2\text {E}\,\left\{ \vartheta _{3}\vartheta _{3}'\left( p_{2}-2\right) ^{2} \mathcal {T}_{n}^{-2}\text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right\} \\&+\,\text {E}\,\left\{ \vartheta _{3}\vartheta _{3}'\text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right\} \\&-\,2\text {E}\,\left\{ \vartheta _{3}\vartheta _{3}'\left( p_{2}-2\right) \mathcal {T}_{n}^{-1}\text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right\} \\&+\,\text {E}\,\left\{ \vartheta _{3}\vartheta _{3}'\left( p_{2}-2\right) ^{2} \mathcal {T}_{n}^{-2}\text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right\} \\= & {} \text {Cov}\, \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{S}} \right) -2\text {E}\,\left\{ \vartheta _{3}\vartheta _{1}'\left\{ 1-\left( p_{2}-2\right) \mathcal {T}_{n}^{-1}\right\} \text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right\} \\&-\,\text {E}\,\left\{ \vartheta _{3}\vartheta _{3}'\left( p_{2}-2\right) ^{2} \mathcal {T}_{n}^{-2}\text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right\} \\&+\,\text {E}\,\left\{ \vartheta _{3}\vartheta _{3}'\text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right\} \text {.}\, \end{aligned}$$
Based on Lemma 2 and the formula for a conditional mean of a bivariate normal, we have
$$\begin{aligned}&\text {E}\,\left\{ \vartheta _{3}\vartheta _{1}'\left\{ 1-\left( p_{2}-2\right) \mathcal {T}_{n}^{-1}\right\} \text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right\} \\&\quad =\text {E}\,\left\{ \text {E}\,\left( \vartheta _{3}\vartheta _{1}'\left\{ 1-\left( p_{2}-2\right) \mathcal {T}_{n}^{-1}\right\} \text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) |\vartheta _{3}\right) \right\} \\&\quad =\text {E}\,\left\{ \vartheta _{3}\text {E}\,\left( \vartheta _{1}'\left\{ 1-\left( p_{2}-2\right) \mathcal {T}_{n}^{-1}\right\} \text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) |\vartheta _{3}\right) \right\} \\&\quad =\text {E}\,\left\{ \vartheta _{3}\left[ -\varvec{\eta }_{11.2}+\left( \vartheta _{3}-\varvec{\delta }\right) \right] '\left\{ 1-\left( p_{2}-2\right) \mathcal {T}_{n}^{-1}\right\} \text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right\} \\&\quad =-\varvec{\eta }_{11.2}\text {E}\,\left( \vartheta _{3}\left\{ 1-\left( p_{2}-2\right) \mathcal {T}_{n}^{-1}\right\} \text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right) \\&\qquad +\,\text {E}\,\left( \vartheta _{3}\vartheta _{3}'\left\{ 1-\left( p_{2}-2\right) \mathcal {T}_{n}^{-1}\right\} \text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right) \\&\qquad -\,\text {E}\,\left( \vartheta _{3}\varvec{\delta }'\left\{ 1-\left( p_{2}-2\right) \mathcal {T}_{n}^{-1}\right\} \text {I}\,\left( \mathcal {T}_{n}\le p_{2}-2\right) \right) \\&\quad =-\varvec{\delta \eta }_{11.2}'\varvec{E}\left( \left\{ 1-\left( p_{2}-2\right) \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \right\} \text {I}\,\left( \chi _{p_{2}+2}^{2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) \\&\qquad +\,\varvec{{\varPhi } }_{*}\text {E}\,\left( \left\{ 1-\left( p_{2}-2\right) \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \right\} \text {I}\,\left( \chi _{p_{2}+2}^{2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) \\&\qquad +\,\varvec{\delta \delta }'\text {E}\,\left( \left\{ 1-\left( p_{2}-2\right) \chi _{p_{2+4}}^{-2}\left( {\varDelta } \right) \right\} \text {I}\,\left( \chi _{p_{2+4}}^{2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) \\&\qquad -\,\varvec{\delta \delta }'\text {E}\,\left( \left\{ 1-\left( p_{2}-2\right) \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \right\} \text {I}\,\left( \chi _{p_{2}+2}^{2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) , \end{aligned}$$
$$\begin{aligned} \text {Cov}\, \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{PS}} \right)= & {} \text {Cov}\, \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{S}} \right) +2\varvec{ \delta \eta }_{11.2}'\text {E}\,\left( \left\{ 1-\left( p_{2}-2\right) \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \right\} \text {I}\,\left( \chi _{p_{2}+2}^{2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) \\&\quad -\,2\varvec{{\varPhi } }_{*}\text {E}\,\left( \left\{ 1-\left( p_{2}-2\right) \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \right\} \text {I}\,\left( \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) \\&\quad -\,2\varvec{\delta \delta }'\text {E}\,\left( \left\{ 1-\left( p_{2}-2\right) \chi _{p_{2}+4}^{-2}\left( {\varDelta } \right) \right\} \text {I}\,\left( \chi _{p_{2}+4}^{2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) \\&\quad +\,2\varvec{\delta \delta }'\text {E}\,\left( \left\{ 1-\left( p_{2}-2\right) \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \right\} \text {I}\,\left( \chi _{p_{2}+2}^{2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) \\&\quad -\,\left( p_{2}-2\right) ^{2}\varvec{{\varPhi } }_{*}\text {E}\,\left( \chi _{p_{2}+2,\alpha }^{-4}\left( {\varDelta } \right) \text {I}\,\left( \chi _{p_{2}+2,\alpha }^{2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) \\&\quad -\,\left( p_{2}-2\right) ^{2}\varvec{\delta \delta }'\text {E}\,\left( \chi _{p_{2}+4}^{-4}\left( {\varDelta } \right) \text {I}\,\left( \chi _{p_{2}+2}^{2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) \\&\quad +\,\varvec{{\varPhi } }_{*}\text {H}\,_{p_{2}+2}\left( p_{2}-2;{\varDelta } \right) +\,\varvec{\delta \delta }'\text {H}\,_{p_{2}+4}\left( p_{2}-2;{\varDelta } \right) \\= & {} \text {Cov}\, \left( \varvec{\widehat{\beta }}_{1}^{\mathrm{S}} \right) +2\varvec{ \delta \eta }_{11.2}'\text {E}\,\left( \left\{ 1-\left( p_{2}-2\right) \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \right\} \text {I}\,\left( \chi _{p_{2}+2}^{2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) \\&\quad +\left( p_{2}-2\right) \sigma ^{2}\varvec{{\tilde{Q}}}_{11}^{-1}\varvec{{\tilde{Q}}} _{12}\varvec{{\tilde{Q}}}_{22.1}^{-1}\varvec{{\tilde{Q}}}_{21}\varvec{{\tilde{Q}}}_{11}^{-1} \\&\quad \times \left[ 2\text {E}\,\left( \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \text {I}\,\left( \chi _{p_{2}+2}^{2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) \right. \\&\quad \left. -\left( p_{2}-2\right) \text {E}\,\left( \chi _{p_{2}+2}^{-4}\left( {\varDelta } \right) \text {I}\,\left( \chi _{p_{2}+2}^{2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) \right] \\&\quad -\,\sigma ^{2}\varvec{{\tilde{Q}}}_{11}^{-1}\varvec{{\tilde{Q}}}_{12}\varvec{{\tilde{Q}}} _{22.1}^{-1}\varvec{{\tilde{Q}}}_{21}\varvec{{\tilde{Q}}}_{11}^{-1}\text {H}\,_{p_{2}+2}\left( p_{2}-2;{\varDelta } \right) \\&\quad +\,\varvec{\delta \delta }'\left[ 2\text {H}\,_{p_{2}+2}\left( p_{2}-2;{\varDelta } \right) -\text {H}\,_{p_{2}+4}\left( p_{2}-2;{\varDelta } \right) \right] \\&\quad -\left( p_{2}-2\right) \varvec{\delta \delta }'\left[ 2\text {E}\,\left( \chi _{p_{2}+2}^{-2}\left( {\varDelta } \right) \text {I}\,\left( \chi _{p_{2}+2}^{2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) \right. \\&\quad -2\text {E}\,\left( \chi _{p_{2}+4}^{-2}\left( {\varDelta } \right) \text {I}\,\left( \chi _{p_{2}+4}^{2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) \\&\quad \left. +\left( p_{2}-2\right) \text {E}\,\left( \chi _{p_{2}+2}^{-4}\left( {\varDelta } \right) \text {I}\,\left( \chi _{p_{2}+2}^{2}\left( {\varDelta } \right) \le p_{2}-2\right) \right) \right] . \end{aligned}$$
\(\square \)
Proof (Theorem 3)
The asymptotic risks of the estimators can be derived by following the definition of ADR
$$\begin{aligned} \text {ADR}\,\left( {\varvec{\beta }}_{1}^{*}\right)= & {} n\text {E}\,\left[ \left( {\varvec{\beta }}_{1}^{*}-{\varvec{\beta }}_{1}\right) ' \mathbf W \left( {\varvec{\beta }}_{1}^{*}-{\varvec{\beta }}_{1}\right) \right] \\ \nonumber= & {} n\text {tr}\,\left[ \mathbf W \text {E}\,\left( {\varvec{\beta }}_{1}^{*}- {\varvec{\beta }}_{1}\right) \left( {\varvec{\beta }}_{1}^{*}- {\varvec{\beta }}_{1}\right) '\right] \\= & {} \text {tr}\,\left( \mathbf W \text {Cov}\, \left( {\varvec{\beta }}_{1}^{*} \right) \right) . \end{aligned}$$
\(\square \)
