Appendix
Proof of Theorem 4.2
Let \(\hat{\varvec{\xi }}=(\hat{\varvec{\xi }}_1^{\top }, \hat{\varvec{\xi }}_2^{\top })^{\top }\) be the solution of \(\sum _{i=1}^n S_i(\varvec{\xi }) = 0\). For parameter \(\varvec{\xi }_2\), we can show under \(H_0\) that \(\sqrt{n}(\hat{\varvec{\xi }}_2-\varvec{0})\) is multivariate normal with mean \(\varvec{0}\) and variance \({\varvec{{\mathcal {I}}}}_{\xi _{22}}\) as \(n\rightarrow \infty \). Hence \(n\hat{\varvec{\xi }}_2^{\top } {\varvec{{\mathcal {I}}}}_{\xi _{22}}^{-1} \hat{\varvec{\xi }}_2\) converges to \(\chi _{p_2}^2\) distribution.
Using the idea of the proof of Result 1 of Lin and Carroll (2001), it can be shown by Taylor expansion that \((\hat{\varvec{\xi }}-\varvec{\xi }) =n^{-1}\varvec{\varOmega }_{\varvec{\xi }} \varvec{S}(\varvec{\xi }_1, \varvec{\xi }_2) +o_p(1)\) and if \(H_0\) is true, we can write
$$\begin{aligned} \hat{\varvec{\xi }}_2 -\varvec{0} = \frac{1}{n} \left( \varvec{\varOmega }^{(1)} \varvec{S}_{\varvec{\xi }_1} (\varvec{\xi }_1,\varvec{0}) + \varvec{\varOmega }_{\varvec{\xi }_{22}} \varvec{S}_{\varvec{\xi }_2} (\varvec{\xi }_1,\varvec{0})\right) +o_p(1) , \end{aligned}$$
where \(\varvec{\varOmega }^{(1)}\) is the lower left of \(p_2\times (q+p_1)\) submatrix of \(\varvec{\varOmega }_{\varvec{\xi }}\). Plug-in the value of \(\hat{\varvec{\xi }}_2\) in the following expression
$$\begin{aligned}&n\hat{\varvec{\xi }}_{2}^{\top } {\varvec{{\mathcal {I}}}}_{{\varvec{\xi }}_{22}}^{-1}\hat{\varvec{\xi }}_{2} \nonumber \\&\quad = n^{-1} {\varvec{S}}_{ {\varvec{\xi }}_1}^{\top }(\varvec{\xi }_1, \varvec{0}) \left( [{\varvec{\varOmega }}^{(1)}]^{\top } {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22}}^{-1} {\varvec{\varOmega }}^{(1)}\right) {\varvec{S}}_{\varvec{\xi }_1}( \varvec{\xi }_1, \varvec{0}) \nonumber \\&\qquad + n^{-1} {\varvec{S}}_{\varvec{\xi }_2}^{\top }(\varvec{\xi }_1, \varvec{0}) {\varvec{\varOmega }}_{\varvec{\xi }_{22}}^{\top } {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22}}^{-1} {\varvec{\varOmega }}^{(1)} {\varvec{S}}_{\varvec{\xi }_1}(\varvec{\xi }_1, \varvec{0}) \nonumber \\&\qquad +n^{-1} {\varvec{S}}_{\varvec{\xi }_1}^{\top }(\varvec{\xi }_1, \varvec{0}) \left( [{\varvec{\varOmega }}^{(1)}]^{\top } {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22}}^{-1} {\varvec{\varOmega }}_{\varvec{\xi }_{22}}\right) {\varvec{S}}_{\varvec{\xi }_2}(\varvec{\xi }_1, \varvec{0}) \nonumber \\&\qquad + n^{-1} {\varvec{S}}_{\varvec{\xi }_2}^{\top }(\varvec{\xi }_1, \varvec{0})\left( {\varvec{\varOmega }}_{\varvec{\xi }_{22}}^{\top } {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22}}^{-1} {\varvec{\varOmega }}_{\varvec{\xi }_{22}}\right) {\varvec{S}}_{\varvec{\xi }_2}( \varvec{\xi }_1, \varvec{0}). \end{aligned}$$
(9)
which has an asymptotic \(\chi _{p_2}^2\) distribution with \(p_2\) degrees of freedom. To show this, assume that \({\hat{\varXi }}_L^*\) be the new version of relation (9) when \( \varvec{\xi }_1\) is replaced by \(\tilde{\varvec{\xi }}_1\). Now the relation (9) differs from \({\hat{\varXi }}_L^*\) by an \(o_p(1)\) term as \(n\rightarrow \infty \). Thus \({\hat{\varXi }}_L^*\) converges in distribution to \(\chi _{p_2}^2\). We know that \({\varvec{S}}_{\varvec{\xi }_1}(\tilde{\varvec{\xi }}_1, \varvec{0})=\varvec{0}\) and this \({\varvec{S}}_{\varvec{\xi }_1}(\tilde{\varvec{\xi }}_1, \varvec{0})\) involves in the first three terms of (9) when \( \varvec{\xi }_1\) is replaced by \(\tilde{\varvec{\xi }}_1\). Thus \({\hat{\varXi }}_L^*= {\hat{\varXi }}_L\) and hence we complete the proof of the theorem.
Under local alternative (6), the following theorem and Lemma facilitates the theoretical derivation of ADB and ADR of the RPSIM, PTSIM, SESIM, and PSESIM estimators in Theorems 4.1.2 and 4.1.3. \(\square \)
Proof of Theorem 4.1.1
$$\begin{aligned} \text{ E }(\varvec{\psi }_1)= & {} \text{ E }\left( \sqrt{n}(\hat{\varvec{\xi }} - \varvec{\xi })\right) = \varvec{0}.\\ \text{ E }(\varvec{\psi }_2)= & {} \text{ E }\left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi })\right) \\= & {} \text{ E }\left( \sqrt{n}(\hat{\varvec{\xi }}_1- \varvec{\xi }_1+ {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} \hat{\varvec{\xi }}_2)\right) ,~~\text{ see } Lawless and Singhal (1978)\\= & {} \varvec{0} + {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} \sqrt{n}\left( \frac{\varvec{\delta }}{\sqrt{n}}\right) ={\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}}\varvec{\delta } =\varvec{\gamma }. \end{aligned}$$
$$\begin{aligned} \text{ E }(\varvec{\psi }_3)= & {} \text{ E }\left( \sqrt{n}(\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})\right) = \text{ E }\left( \sqrt{n}(\hat{\varvec{\xi }} - \varvec{\xi })-\sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi })\right) = - \varvec{\gamma }.\\ \text{ Var }(\varvec{\psi }_1)= & {} \text{ Var }\left( \sqrt{n}(\hat{\varvec{\xi }} - \varvec{\xi })\right) = {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11.2}}^{-1} ={\mathcal {A}}_{11}.\\ \text{ Var }(\varvec{\psi }_2)= & {} \text{ Var }\left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi })\right) =\text{ Var }\left( \sqrt{n}(\hat{\varvec{\xi }}_1- \varvec{\xi }_1+ {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} \hat{\varvec{\xi }}_2)\right) \\= & {} \text{ Var }\left( \sqrt{n}\left( \hat{\varvec{\xi }}_1- \varvec{\xi }_1\right) \right) + {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} \text{ Var }\left( \sqrt{n} \hat{\varvec{\xi }}_2\right) {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{21}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} \\&+ 2 \text{ Cov }\left( \sqrt{n}\left( \hat{\varvec{\xi }}_1- \varvec{\xi }_1\right) , {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} \sqrt{n} \hat{\varvec{\xi }}_2\right) \\= & {} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11.2}}^{-1} - {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22.1}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{21}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} = {\mathcal {A}}_{22}. \end{aligned}$$
$$\begin{aligned} \text{ Var }(\varvec{\psi }_3)= & {} \text{ Var }\left( \sqrt{n}(\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})\right) = \text{ Var }\left( \sqrt{n}(\hat{\varvec{\xi }} - \varvec{\xi })-\sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi })\right) \\= & {} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22.1}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{21}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} = {\mathcal {A}}_{33}.\\ \text{ Cov }(\varvec{\psi }_1, \varvec{\psi }_2)= & {} \text{ Cov } \left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi }), \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi })\right) \\= & {} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11.2}}^{-1} - {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22.1}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{21}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} = {\mathcal {A}}_{12} = {\mathcal {A}}_{21}^{\top }.\\ \text{ Cov }(\varvec{\psi }_1, \varvec{\psi }_3)= & {} \text{ Cov } \left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi }), \sqrt{n}(\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})\right) \\= & {} \text{ Cov } \left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi }), \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi })-\sqrt{n} (\tilde{\varvec{\xi }} - \varvec{\xi })\right) \\= & {} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22.1}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{21}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1}= {\mathcal {A}}_{13} = {\mathcal {A}}_{31}^{\top } \end{aligned}$$
$$\begin{aligned} \text{ Cov }(\varvec{\psi }_2, \varvec{\psi }_3)= & {} \text{ Cov } \left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi }), \sqrt{n}(\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})\right) \\= & {} \text{ Cov } \left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi }), \sqrt{n}(\hat{\varvec{\xi }}-\varvec{\xi }) - \sqrt{n}(\tilde{\varvec{\xi }}-\varvec{\xi })\right) \\= & {} \text{ Cov } \left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi }), \sqrt{n}(\hat{\varvec{\xi }}-\varvec{\xi })\right) - \text{ Cov } \left( \sqrt{n}(\tilde{\varvec{\xi }} - \varvec{\xi }), \sqrt{n}(\tilde{\varvec{\xi }}-\varvec{\xi })\right) \\= & {} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11.2}}^{-1} - {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22.1}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{21}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} -{\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11.2}}^{-1} \\&\qquad + {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{22.1}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{21}} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1}=\varvec{0}. \end{aligned}$$
\(\square \)
Lemma
Let \(\varvec{X}\sim \text{ n } (\varvec{\mu }, \varvec{{\mathcal {I}}})\), where \(\varvec{{\mathcal {I}}}\) is a nonnegative definite matrix. Also let \(\varvec{Q}\) be symmetric and positive definite matrix such that \(\varvec{Q}^{1/2} \varvec{{\mathcal {I}}} \varvec{Q}^{1/2}\) is an idempotent matrix, and \(\varvec{Q} \varvec{{\mathcal {I}}} \varvec{Q}\varvec{\mu }=\varvec{Q}\varvec{\mu }\). Then for all \(\varphi \), Borel measurable and real-valued integrable function
$$\begin{aligned}&1 \text{ E }\left( \varphi \left( \varvec{X}^{\top } \varvec{Q} \varvec{{\mathcal {I}}} \varvec{Q}\varvec{X}\right) \varvec{X}\right) = \varvec{\theta }\text{ E }\left( \varphi \left( \chi _{p+2}^2(\varvec{\mu }^{\top } \varvec{Q} \varvec{\mu }) \right) \right) \\&2 \text{ E }\left( \varphi \left( \varvec{X}^{\top } \varvec{Q} \varvec{{\mathcal {I}}} \varvec{Q}\varvec{X}\right) \varvec{X}^{\top }\varvec{M}\varvec{X}\right) = \text{ E }\left( \varphi \left( \chi _{p+2}^2(\varvec{\mu }^{\top } \varvec{Q} \varvec{\mu }) \right) \right) \text{ tr }(\varvec{A\varvec{{\mathcal {I}}}})\\&\qquad +\text{ E }\left( \varphi \left( \chi _{p+4}^2(\varvec{\mu }^{\top } \varvec{Q} \varvec{\mu }) \right) \right) \varvec{\mu }^{\top } \varvec{M} \varvec{\mu }, \end{aligned}$$
where \(\varvec{M}\) is nonnegative definite matrix.
The outline of the proof of this lemma is given in Nkurunziza et al. (2013).
Proof of Theorem 4.1.2
In this proof, we derive the bias expressions of the proposed estimators. It is obvious that \(\text{ ADB }(\hat{\varvec{\xi }})=\varvec{0} \). The ADB of RPSIM (\(\tilde{\varvec{\xi }}\)), PTSIM (\(\hat{\varvec{\xi }}_p\)), SESIM(\(\hat{\varvec{\xi }}_S\)), and PSESIM (\(\hat{\varvec{\xi }}_{S+}\)) estimators are as follows:
$$\begin{aligned} \text{ ADB }(\tilde{\varvec{\xi }})&= \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} (\tilde{\varvec{\xi }} - \varvec{\xi }) \right) = \text{ E }(\varvec{\psi }_2) = -{\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{11}}^{-1} {\varvec{{\mathcal {I}}}}_{\varvec{\xi }_{12}} \varvec{\delta } = \varvec{\gamma }\\ \text{ ADB }(\hat{\varvec{\xi }}_{P})&= \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} (\hat{\varvec{\xi }}_{P} - \varvec{\xi }) \right) = \varvec{0}- \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} I\left( {\hat{\varXi }}_L \le \chi ^2_{\kappa +2, \alpha }\right) \right) \left( \hat{\varvec{\xi }} - \tilde{\varvec{\xi }}\right) \\&= - \text{ E } \left( \lim _{n\rightarrow \infty } I\left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }, \varDelta \right) \varvec{\psi }_3 \right) = \varvec{\gamma } H_{\kappa +4} \left( \chi ^2_{\kappa +2, \alpha }, \varDelta \right) \\ \text{ ADB }(\hat{\varvec{\xi }}_{S})&= \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} (\hat{\varvec{\xi }}_{S} - \varvec{\xi }) \right) = - \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} (\kappa {\hat{\varXi }}_L^{-1}(\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})) \right) \\&= -\kappa \text{ E } \left( \lim _{n\rightarrow \infty } \varvec{\psi }_3 {\hat{\varXi }}_L^{-1}\right) =\kappa \varvec{\gamma } \text{ E }(Z_1^{-1})\\ \text{ ADB }(\hat{\varvec{\xi }}_{S+})&= \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} (\hat{\varvec{\xi }}_{S+} - \varvec{\xi }) \right) \\&= \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n}(\hat{\varvec{\xi }}_{S} - \varvec{\xi }) - \sqrt{n} \left( 1-\kappa {\hat{\varXi }}_L^{-1}\right) I\left( {\hat{\varXi }}_L< \kappa \right) (\hat{\varvec{\xi }}-\tilde{\varvec{\xi }}) \right) \\&= \text{ ADB }(\hat{\varvec{\xi }}_{S}) - \text{ E } \left( \lim _{n\rightarrow \infty } \varvec{\psi }_3 \left( 1-\kappa {\hat{\varXi }}_L^{-1}\right) I\left( {\hat{\varXi }}_L< \kappa \right) \right) \\&= \text{ ADB }(\hat{\varvec{\xi }}_{S})+ \varvec{\gamma } \text{ E } \left( I\left( {\hat{\varXi }}_L< \kappa \right) \right) - \varvec{\gamma }\kappa \text{ E }\left( \varvec{\psi }_3 {\hat{\varXi }}_L^{-1}I\left( {\hat{\varXi }}_L< \kappa \right) \right) \\&= \text{ ADB }(\hat{\varvec{\xi }}_{S})+ \varvec{\gamma } H_{\kappa +4}(\kappa ,\varDelta ) - \varvec{\gamma }\kappa \text{ E }\left( Z_1^{-1} I\left( Z_1< \kappa \right) \right) \end{aligned}$$
Proof of Theorem 4.1.3
Based on the definition of ADR function, it is necessary to derive the asymptotic covariance matrices for the four estimators. The covariance matrix of any estimator \(\hat{\varvec{\xi }}^*\) is defined as:
$$\begin{aligned} \text{ Cov } (\hat{\varvec{\xi }}^*) = \text{ E }\left( \lim _{n\rightarrow \infty } n (\hat{\varvec{\xi }}^*-\varvec{\xi }) (\hat{\varvec{\xi }}^*-\varvec{\xi })^{\top } \right) . \end{aligned}$$
First, we will start deriving the covariance matrices of the UPSIM and RPSIM:
$$\begin{aligned} \text{ Cov } (\hat{\varvec{\xi }})= & {} \text{ E }\left( \lim _{n\rightarrow \infty } \sqrt{n} (\hat{\varvec{\xi }} -\varvec{\xi }) \sqrt{n}( \hat{\varvec{\xi }} -\varvec{\xi })^{\top } \right) = \text{ E }(\varvec{\psi } \varvec{\psi } ^{\top }) = {\varvec{{\mathcal {A}}}}_{12} ={\varvec{{\mathcal {I}}}}^{-1}_{11.2}.\\ \text{ Cov }( \tilde{\varvec{\xi }})= & {} \text{ E }\left( \lim _{n\rightarrow \infty } \sqrt{n}(\tilde{\varvec{\xi }} -\varvec{\xi }) \sqrt{n}(\tilde{\varvec{\xi }} -\varvec{\xi })^{\top }\right) = \text{ E }(\varvec{\psi }_2 \varvec{\psi }_2^{\top })= {\varvec{{\mathcal {A}}}}_{12} +\varvec{\gamma } \varvec{\gamma }^{\top }. \end{aligned}$$
Second, we derive the covariance matrix of the pretest estimator:
$$\begin{aligned} \text{ Cov }( \hat{\varvec{\xi }}_{P})= & {} \text{ E }\left( \lim _{n\rightarrow \infty } \sqrt{n}(\hat{\varvec{\xi }}_{P} - \varvec{\xi }) \sqrt{n} (\hat{\varvec{\xi }}_{P} - \varvec{\xi })^{\top } \right) \\= & {} \text{ E } \left( \varvec{\psi }_1 \varvec{\psi }_1^{\top } + \varvec{\psi }_3 \varvec{\psi }_3^{\top } \lim _{n\rightarrow \infty } I \left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) -2 \varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } I \left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) \\= & {} {\varvec{{\mathcal {A}}}_{11}} + {\varvec{{\mathcal {A}}}_{13}} H_{\kappa + 4}\left( \chi ^2_{\kappa + 2,\alpha }, \varDelta \right) + \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 6}\left( \chi ^2_{\kappa + 2,\alpha }, \varDelta \right) \\&- 2 \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } I\left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) . \end{aligned}$$
Consider the fourth term:
$$\begin{aligned}&\text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } I \left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) =\text{ E } \left( \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } I \left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) | \varvec{\psi }_3\right) \right) \\&\quad = \text{ E } \left( \varvec{\psi }_3 \left( \varvec{\psi }_3^{\top } - \text{ E }\left( \varvec{\psi }_3\right) ^{\top }\right) \lim _{n\rightarrow \infty } I \left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) \\&\quad = \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_3^{\top } \lim _{n\rightarrow \infty } I \left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) -\text{ E } \left( \varvec{\psi }_3 \lim _{n\rightarrow \infty } I \left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) \text{ E }\left( \varvec{\psi }_3\right) ^{\top } \\&\quad = {\varvec{{\mathcal {A}}}_{13}} H_{\kappa +4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) + \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa +6}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) \\&\qquad - \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa +4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) \\&\quad = {\varvec{{\mathcal {A}}}_{13}} H_{\kappa + 4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) + \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 6}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) - \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) ,\\ \text{ Cov }( \hat{\varvec{\xi }}_{P})= & {} {\varvec{{\mathcal {I}}}}^{-1}_{11.2}+ \lim _{n\rightarrow \infty } \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_3^{\top } I\left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) -2 \lim _{n\rightarrow \infty } \text{ E } \left( \varvec{\psi }_3 \varvec{\psi } ^{\top } I\left( {\hat{\varXi }}_L \le \chi ^2_{p_2, \alpha }\right) \right) \\&\quad = {\varvec{{\mathcal {I}}}}^{-1}_{11.2} + {\varvec{{\mathcal {A}}}_{13}} H_{\kappa + 4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) + \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 6}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) \\&\qquad -2 \left( {\varvec{{\mathcal {A}}}_{13}} H_{\kappa + 4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) + \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 6}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) \right. \\&\left. -\varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) \right) \\&\quad = {\varvec{{\mathcal {I}}}}^{-1}_{11.2} - {\varvec{{\mathcal {A}}}_{13}} H_{\kappa + 4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) - \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 6}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) \\&\qquad + 2 \varvec{\gamma } \varvec{\gamma }^{\top } H_{\kappa + 4}\left( \chi ^2_{\kappa + 2, \alpha }, \varDelta \right) . \end{aligned}$$
Third, we derive the covariance matrices of the shrinkage and positive shrinkage estimators:
$$\begin{aligned} \text{ Cov } ( \hat{\varvec{\xi }}_{S})= & {} \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n}(\hat{\varvec{\xi }}_{S} - \varvec{\xi }) \sqrt{n} (\hat{\varvec{\xi }}_{S} - \varvec{\xi })^{\top } \right) \\= & {} \text{ E } \Bigg (\lim _{n\rightarrow \infty } \sqrt{n} \left( \hat{\varvec{\xi }} - \varvec{\xi } - \kappa {\hat{\varXi }}_L^{-1}(\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})\right) \sqrt{n} \left( \hat{\varvec{\xi }} - \varvec{\xi } - \kappa {\hat{\varXi }}_L^{-1}(\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})\right) ^{\top } \Bigg ) \\= & {} \text{ E } (\varvec{\psi }_1 \varvec{\psi }_1^{\top }) + \kappa ^2 \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_3^{\top } \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-2}\right) -2 \kappa \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1}\right) \\= & {} {\varvec{{\mathcal {I}}}}^{-1}_{11.2} + {\varvec{{\mathcal {A}}}_{13}} \text{ E }\left( Z_1^{-2}\right) + \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E }\left( Z_2^{-2}\right) -2 \kappa \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1}\right) . \end{aligned}$$
Consider the last term:
$$\begin{aligned}&\text{ E } (\varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1})=\text{ E } \left( \text{ E } (\varvec{\psi }_3 \varvec{\psi }_1^{\top } \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1} | \varvec{\psi }_3)\right) \\&\quad = \text{ E } \left( \varvec{\psi }_3 \text{ E } (\varvec{\psi }_1^{\top } | \varvec{\psi }_3) \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1}\right) + \text{ E } \left( \varvec{\psi }_3 \left( \varvec{\psi }_3^{\top } - \text{ E }(\varvec{\psi }_3\right) ^{\top }) \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1}\right) \\&\quad = \text{ E } \left( \varvec{\psi }_3 \varvec{\psi }_3^{\top } \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1}\right) - \text{ E } \left( \varvec{\psi }_3 \lim _{n\rightarrow \infty } {\hat{\varXi }}_L^{-1}\right) \text{ E }\left( \varvec{\psi }_3\right) ^{\top } \\&\quad = {\varvec{{\mathcal {A}}}_{13}} \text{ E } \left( \chi ^{-2}_{p_2 + 2}(\varDelta )\right) + \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E } \left( \chi ^{-2}_{p_2 + 4}(\varDelta )\right) - \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E } \left( \chi ^{-2}_{p_2 + 2}(\varDelta )\right) \\&\quad = {\varvec{{\mathcal {A}}}_{13}} \text{ E } \left( Z_1^{-1}\right) + \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E } \left( Z_2^{-1}\right) - \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E } \left( Z_1^{-1}\right) . \end{aligned}$$
Hence
$$\begin{aligned} \text{ Cov } ( \hat{\varvec{\xi }}_{S})= & {} {\varvec{{\mathcal {I}}}}^{-1}_{11.2} + \kappa ^2 \left( {\varvec{{\mathcal {A}}}_{13}} \text{ E } (Z_1^{-2}) + \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E }(Z_2^{-2}) \right) \\&-2 \kappa \left( {\varvec{{\mathcal {A}}}_{13}} \text{ E }(Z_1^{-1}) + \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E } (Z_2^{-1}) - \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E } (Z_1^{-1})\right) \\= & {} {\varvec{{\mathcal {I}}}}^{-1}_{11.2} + \left( \kappa ^2 \text{ E } \left( Z_1^{-2}\right) -2 \kappa \text{ E } (Z_1^{-1}) \right) {\varvec{{\mathcal {A}}}_{13}}\\&+ \left( \kappa ^2 \text{ E } (Z_2^{-2}) +2 \kappa \text{ E } (Z_1^{-1}) -2 \kappa \text{ E } (Z_2^{-1}) \right) \varvec{\gamma } \varvec{\gamma }^{\top }. \end{aligned}$$
Let \(F_m(\varDelta )=\left( 1-\kappa {\hat{\varXi }}_L^{-1}\right) ^m I \left( {\hat{\varXi }}_L < \kappa \right) \), where \(m=1,2\)
$$\begin{aligned} \text{ Cov } (\hat{\varvec{\xi }}_{S+})= & {} \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} (\hat{\varvec{\xi }}_{S+} - \varvec{\xi }) \sqrt{n} (\hat{\varvec{\xi }}_{S+} - \varvec{\xi })^{\top }\right) , \\= & {} \text{ E } \left( \lim _{n\rightarrow \infty } \sqrt{n} (\hat{\varvec{\xi }}_{S} - \varvec{\xi }) \sqrt{n} (\hat{\varvec{\xi }}_{S} - \varvec{\xi })^{\top }\right) \\&+ \text{ E } \left( \lim _{n\rightarrow \infty } F_2(\varDelta ) \sqrt{n} (\hat{\varvec{\xi }} - \tilde{\varvec{\xi }}) \sqrt{n} (\hat{\varvec{\xi }} - \tilde{\varvec{\xi }})^{\top }\right) \\&-2 \text{ E } \left( \lim _{n\rightarrow \infty } F_1(\varDelta ) \sqrt{n} (\hat{\varvec{\xi }} - \tilde{\varvec{\xi }}) \sqrt{n}(\hat{\varvec{\xi }}_{S} - \varvec{\xi })^{\top }\right) \\= & {} \text{ Cov } ( \hat{\varvec{\xi }}_{S}) + \text{ E } \left( \lim _{n\rightarrow \infty } F_2(\varDelta ) \varvec{\psi }_3 \varvec{\psi }_2^{\top }\right) \\- & {} 2 \text{ E } \left( \lim _{n\rightarrow \infty } F_1(\varDelta ) \varvec{\psi }_3 \left( \varvec{\psi }_2^{\top } + \left( 1-\kappa {\hat{\varXi }}_L^{-1}\right) \varvec{\psi }_3^{\top }\right) \right) , \\= & {} \text{ Cov } ( \hat{\varvec{\xi }}_{S}) - \text{ E } \left( \lim _{n\rightarrow \infty }F_2(\varDelta )\varvec{\psi }_3 \varvec{\psi }_3^{\top }\right) -2 \text{ E } \left( \lim _{n\rightarrow \infty } F_1(\varDelta ) \varvec{\psi }_3 \varvec{\psi }_2^{\top }\right) . \end{aligned}$$
\(\square \)
Consider the second term:
$$\begin{aligned}&- \text{ E } \left( \lim _{n\rightarrow \infty } F_2(\varDelta )\varvec{\psi }_3 \varvec{\psi }_3^{\top }\right) = - \text{ E } \left( \lim _{n\rightarrow \infty } \left( 1-\kappa {\hat{\varXi }}_L^{-1}\right) ^2 I \left( {\hat{\varXi }}_L< \kappa \right) \varvec{\psi }_3 \varvec{\psi }_3^{\top }\right) \\&\quad = - {\varvec{{\mathcal {A}}}_{13}} \text{ E } \left( I \left( Z_1< \kappa \right) \left( 1 - \kappa Z_1^{-1} \right) ^2\right) - \varvec{\gamma } \varvec{\gamma }^{\top } \text{ E } \left( I \left( Z_2 < \kappa \right) \left( 1 - \kappa Z_2^{-1} \right) ^2\right) . \end{aligned}$$
Consider the third term:
$$\begin{aligned}&-2 \text{ E } \left( \lim _{n\rightarrow \infty } F_1(\varDelta ) \varvec{\psi }_3 \varvec{\psi }_2^{\top }\right) = -2 \text{ E } \left( \lim _{n\rightarrow \infty } \varvec{\psi }_3 \text{ E } \left( F_1(\varDelta ) \varvec{\psi }_2^{\top } | \varvec{\psi }_3 \right) \right) \\&\quad = -2 \text{ E } \left( \lim _{n\rightarrow \infty } \varvec{\psi }_3 \left( \text{ E } \left( \varvec{\psi }_2^{\top }\right) + \text{ cov } \left( \varvec{\psi }_3, \varvec{\psi }_2\right) \left( \varvec{\psi }_3 - \text{ E }\left( \varvec{\psi }_3\right) \right) \right) F_1(\varDelta )\right) \\&\quad = -2 \text{ E } \left( \lim _{n\rightarrow \infty } \varvec{\psi }_3 \text{ E } \left( \varvec{\psi }_2^{\top }\right) F_1(\varDelta ) + \varvec{0}\right) \\&\quad = -2 \text{ E } \left( \lim _{n \rightarrow \infty } \varvec{\psi }_3 I \left( {\hat{\varXi }}_L< \kappa \right) - \kappa {\hat{\varXi }}_L^{-1} \varvec{\psi }_3 I \left( {\hat{\varXi }}_L< \kappa \right) \right) \text{ E } \left( \varvec{\psi }_2^{\top }\right) \\&\quad = 2 H_{\kappa + 4} (\kappa , \varDelta ) \varvec{\gamma } \varvec{\gamma }^{\top } - 2 \kappa \text{ E } \left( Z_1^{-1} I \left( Z_1< \kappa \right) \right) \varvec{\gamma } \varvec{\gamma }^{\top } \\&\quad = \left( 2 H_{\kappa + 4} (\kappa , \varDelta ) - 2 \kappa \text{ E } \left( Z_1^{-1} I \left( Z_1< \kappa \right) \right) \right) \varvec{\gamma } \varvec{\gamma }^{\top }. \end{aligned}$$
Finally,
$$\begin{aligned} \text{ Cov } \left( \hat{\varvec{\xi }}_{S+}\right)= & {} \text{ Cov } \left( \hat{\varvec{\xi }}_{S}\right) - \text{ E } \left( \left( 1 - \kappa Z_1^{-1}\right) ^2 I \left( Z_1< \kappa \right) \right) {\varvec{{\mathcal {A}}}_{13}} \\&+ \left( 2 H_{\kappa + 4} (\kappa , \varDelta ) - 2 \kappa \text{ E } \left( Z_1^{-1} I \left( Z_1< \kappa \right) \right) \right. \\&- \left. \text{ E } \left( \left( 1 - \kappa Z_2^{-1} \right) ^2 I \left( Z_2 < \kappa \right) \right) \right) \varvec{\gamma } \varvec{\gamma }^{\top }. \end{aligned}$$
The ADR expressions in Theorem 4.4 now follow from (8) which completes the proof.