Appendix: proofs of theorems
Proof of Lemma 1
The estimated control calibrated weights can be expressed as \(w_{i}=d_{i}^{A}g_{i}\), where \(g_{i}=1/\left( 1-q_{i}^{A}x_{i}^{\text {T}}\lambda \right)\) by the standard Lagrange multiplier method. The Lagrange multiplier \(\lambda =\left( \lambda _{1},\lambda _{2},\ldots ,\lambda _{p} \right) ^{\text {T}}\) can be obtained by solving the following equation
$$\begin{aligned} \sum \limits _{i\in {s_{A}}}\frac{d_{i}^{A}x_{i}^{\text {T}}}{1-q_{i}^{A}x_{i}^{\text {T}}\lambda }-\sum \limits _{i\in {s_{B}}}d_{i}^{B}x_{i}^{\text {T}}=0. \end{aligned}$$
(A1)
Define \(d_{i}^{A}x_{i}^{\text {T}}=d_{i}^{A}x_{i}^{\text {T}}\left( 1-q_{i}^{A}x_{i}^{\text {T}}\lambda +q_{i}^{A}x_{i}^{\text {T}}\lambda \right)\) and substitute it into (A1), we can obtain
$$\begin{aligned}&\sum \limits _{i\in {s_{A}}}\frac{d_{i}^{A}x_{i}^{\text {T}}\left( 1-q_{i}^{A}x_{i}^{\text {T}}\lambda +q_{i}^{A}x_{i}^{\text {T}}\lambda \right) }{1-q_{i}^{A}x_{i}^{\text {T}}\lambda }-\sum \limits _{i\in {s_{B}}}d_{i}^{B}x_{i}^{\text {T}}=0\nonumber ,\\&\sum \limits _{i\in {s_{A}}}d_{i}^{A}x_{i}^{\text {T}}+\sum \limits _{i\in {s_{A}}}\frac{\lambda ^{\text {T}}d_{i}^{A}q_{i}^{A}x_{i}x_{i}^{\text {T}}}{1-q_{i}^{A}x_{i}^{\text {T}}\lambda }-\sum \limits _{i\in {s_{B}}}d_{i}^{B}x_{i}^{\text {T}}=0\nonumber ,\\&\lambda ^{\text {T}}\sum \limits _{i\in {s_{A}}}\frac{d_{i}^{A}q_{i}^{A}x_{i}x_{i}^{\text {T}}}{1-\lambda ^{\text {T}}q_{i}^{A}x_{i}}=\sum \limits _{i\in {s_{B}}}d_{i}^{B}x_{i}^{\text {T}}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}x_{i}^{\text {T}}. \end{aligned}$$
(A2)
It follows from (A2) that
$$\begin{aligned} \frac{\lambda ^{*\text {T}}}{1+\lambda ^{*\text {T}}u^{*}}\sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}x_{i}x_{i}^{\text {T}}\le {{\mathbf {R}}}, \end{aligned}$$
(A3)
where \(\lambda ^{*}=(\left| \lambda _{1}\right| , \cdots , \left| \lambda _{p}\right| )^{\text {T}}\), \(u^{*}=\left( \max \limits _{i\in {s_{A}}}{|q_{i}^{A}x_{i1}|},\max \limits _{i\in {s_{A}}}{|q_{i}^{A}x_{i2}|},\ldots ,\max \limits _{i\in {s_{A}}}{|q_{i}^{A}x_{ip}|} \right) ^{\text {T}}=o_{p}\left( n_{A}^{1/2} \right)\) by the condition (A1) and \({\mathbf {R}}=\left( \left| \sum \limits _{i\in {s_{B}}}d_{i}^{B}x_{i1}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}x_{i1}\right| ,\ldots ,\left| \sum \limits _{i\in {s_{B}}}d_{i}^{B}x_{ip}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}x_{ip}\right| \right)\). The condition (A2) implies \(\mathbf {{\widehat{T}}}^{X}_{A}={\mathbf {T}}^{X}+O_{p}\left( Nn_{A}^{-1/2} \right)\), where \(\widehat{{\mathbf {T}}}^{X}_{A}=\left( \widehat{{\mathbf {T}}}^{X}_{A1},\widehat{{\mathbf {T}}}^{X}_{A2},\ldots ,\widehat{{\mathbf {T}}}^{X}_{Ap} \right) =\left( \sum \limits _{i\in {s_{A}}}d_{i}^{A}x_{i1},\sum \limits _{i\in {s_{A}}}d_{i}^{A}x_{i2},\ldots ,\sum \limits _{i\in {s_{A}}}d_{i}^{A}x_{ip} \right)\), and the condition (A3) implies \(\mathbf {{\widehat{T}}}^{X}_{B}={\mathbf {T}}^{X}+O_{p}\left( Nn_{B}^{-1/2} \right)\). Furthermore, we can obtain \(\widehat{{\mathbf {T}}}^{X}_{B}-\widehat{{\mathbf {T}}}^{X}_{A}=O_{p}\left( Nn_{*}^{-1/2} \right)\), where \(n_{*}=\min \left( n_{A},n_{B} \right)\). Under conditions (A1)-(A4), we have \(\sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}x_{i}x_{i}^{\text {T}}=O\left( N \right)\), \(\lambda ^{\text {T}}=O_{p}\left( n_{*}^{-1/2} \right)\) and \(\max \limits _{i\in {s_{A}}}|\lambda ^{\text {T}}q_{i}^{A}x_{i}|=o_{p}(1)\). Furthermore, we can obtain
$$\begin{aligned}&\left( \lambda ^{ECGPEL} \right) ^{\text {T}}\nonumber \\&\quad =\left( \sum \limits _{i\in {s_{B}}}d_{i}^{B}x_{i}^{\text {T}}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}x_{i}^{\text {T}} \right) \left( \sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}x_{i}x_{i}^{\text {T}} \right) ^{-1}\left( 1+o_{p}\left( 1 \right) \right) \nonumber \\&\quad =\left( \sum \limits _{i\in {s_{B}}}d_{i}^{B}x_{i}^{\text {T}}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}x_{i}^{\text {T}} \right) \left( \sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}x_{i}x_{i}^{\text {T}} \right) ^{-1} \nonumber \\&\qquad +\left( \sum \limits _{i\in {s_{B}}}d_{i}^{B}x_{i}^{\text {T}}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}x_{i}^{\text {T}} \right) \left( \sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}x_{i}x_{i}^{\text {T}} \right) ^{-1}o_{p}\left( 1 \right) \nonumber \\&\quad =\left( \sum \limits _{i\in {s_{B}}}d_{i}^{B}x_{i}^{\text {T}}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}x_{i}^{\text {T}} \right) \left( \sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}x_{i}x_{i}^{\text {T}} \right) ^{-1}\nonumber \\&\qquad +O_{p}\left( Nn_{*}^{-1/2} \right) O_{p}\left( 1/N \right) o_{p}\left( 1 \right) \nonumber \\&\quad =\left( \sum \limits _{i\in {s_{B}}}d_{i}^{B}x_{i}^{\text {T}}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}x_{i}^{\text {T}} \right) \left( \sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}x_{i}x_{i}^{\text {T}} \right) ^{-1}+o_{p}\left( n_{*}^{-1/2} \right) . \nonumber \end{aligned}$$
Then, we have
$$\begin{aligned} \lambda ^{ECGPEL}&=\left( \sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}x_{i}x_{i}^{\text {T}} \right) ^{-1}\left( \sum \limits _{i\in {s_{B}}}d_{i}^{B}x_{i}^{\text {T}}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}x_{i}^{\text {T}} \right) ^{\text {T}}+o_{p}\left( n_{*}^{-1/2} \right) \nonumber \\&=\left( {\mathbf {X}}_{A}^{\text {T}}{\mathbf {D}}^{A}{\mathbf {Q}}^{A}{\mathbf {X}}_{A} \right) ^{-1}\left[ \left( {\mathbf {d}}^{B} \right) ^{\text {T}}{\mathbf {X}}_{B}-\left( {\mathbf {d}}^{A} \right) ^{\text {T}}{\mathbf {X}}_{A} \right] ^{\text {T}}+o_{p}\left( n_{*}^{-1/2} \right) , \end{aligned}$$
(A4)
where \({\mathbf {Q}}^{A}\) is a \(n_{A}\times {n_{A}}\) diagonal matrix in which the diagonal elements is \(q_{i}^{A}\) in non-probability sample, \({\mathbf {d}}^{B}=\left( d_{1}^{B},d_{2}^{B},\ldots ,d_{n_{B}}^{B} \right) ^{\text {T}}\), \({\mathbf {X}}_{B}=\left( x_{1},x_{2},\ldots ,x_{n_{B}} \right) ^{\text {T}}\), \(x^{\text {T}}_{i}=(x_{i1}, x_{i2}, \cdots , x_{ip})\), \(i=1, 2, \cdots , n_{B}\). By Taylor’s expansion, we can obtain \(g_{i}=1/\left( 1-q_{i}^{A}x_{i}^{\text {T}}\lambda \right) =1+q_{i}^{A}x_{i}^{\text {T}}\lambda +o_{p}\left( n_{A}^{-1/2} \right) \doteq 1+q_{i}^{A}x_{i}^{\text {T}}\lambda\). Then, the ECGPEL estimator of estimated control calibrated weights is given by
$$\begin{aligned}&{\mathbf {W}}^{ECGPEL}\nonumber \\&\quad = {\mathbf {D}}^{A}\left[ {\mathbf {E}}+{\mathbf {Q}}^{A}{\mathbf {X}}_{A}\lambda ^{ECGPEL}+o_{p}\left( n_{A}^{-1/2} \right) \right] \nonumber \\&\quad ={\mathbf {d}}^{A}+{\mathbf {D}}^{A}{\mathbf {Q}}^{A}{\mathbf {X}}_{A}\left( {\mathbf {X}}_{A}^{\text {T}}{\mathbf {D}}^{A}{\mathbf {Q}}^{A}{\mathbf {X}}_{A} \right) ^{-1}\left[ \left( {\mathbf {d}}^{B} \right) ^{\text {T}}{\mathbf {X}}_{B}-\left( {\mathbf {d}}^{A} \right) ^{\text {T}}{\mathbf {X}}_{A} \right] ^{\text {T}}+o_{p}\left( n_{*}^{-1/2} \right) \nonumber \\&\qquad +o_{p}\left( n_{A}^{-1/2} \right) \nonumber \\&\quad ={\mathbf {d}}^{A}+{\mathbf {D}}^{A}{\mathbf {Q}}^{A}{\mathbf {X}}_{A}\left(
{\mathbf {X}}_{A}^{\text {T}}{\mathbf {D}}^{A}{\mathbf {Q}}^{A}{\mathbf {X}}_{A} \right) ^{-1}\left[ \left( {\mathbf {d}}^{B} \right) ^{\text {T}}{\mathbf {X}}_{B}-\left( {\mathbf {d}}^{A} \right) ^{\text {T}}{\mathbf {X}}_{A} \right] ^{\text {T}}+o_{p}\left( n_{*}^{-1/2} \right) , \end{aligned}$$
(A5)
where \({\mathbf {E}}={\mathbf {1}}_{(n_{A} \times 1)}\). Therefore, the ECGPEL estimator of the population mean is
$$\begin{aligned}&\widehat{\overline{{\mathbf {Y}}}}^{ECGPEL}\nonumber \\&\quad ={\widehat{N}}^{-1}\left( {\mathbf {W}}^{ECGPEL} \right) ^{\text {T}}{\mathbf {y}}\nonumber \\&\quad ={\widehat{N}}^{-1}\left( {\mathbf {d}}^{A} \right) ^{\text {T}}{\mathbf {y}}+{\widehat{N}}^{-1}\left[ \left( {\mathbf {d}}^{B} \right) ^{\text {T}}{\mathbf {X}}_{B}-\left( {\mathbf {d}}^{A} \right) ^{\text {T}}{\mathbf {X}}_{A} \right] \left( {\mathbf {X}}_{A}^{\text {T}}{\mathbf {Q}}^{A}{\mathbf {D}}^{A}{\mathbf {X}}_{A} \right) ^{-1}{\mathbf {X}}_{A}^{\text {T}}{\mathbf {Q}}^{A}{\mathbf {D}}^{A}{\mathbf {y}}\nonumber \\&\qquad +o_{p}\left( n_{*}^{-1/2} \right) \nonumber \\&\quad =\widehat{\overline{{\mathbf {Y}}}}^{PHT}+{\widehat{N}}^{-1}\left( \sum \limits _{i\in {s_{B}}}d_{i}^{B}x_{i}^{\text {T}}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}x_{i}^{\text {T}} \right) \left( \sum \limits _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}x_{i}x_{i}^{\text {T}} \right) ^{-1}\sum \limits _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}x_{i}y_{i}\nonumber \\&\qquad +o_{p}\left( n_{*}^{-1/2} \right) \nonumber \\&\quad =\widehat{\overline{{\mathbf {Y}}}}^{ECGREG}+o_{p}\left( n_{*}^{-1/2} \right) , \end{aligned}$$
(A6)
where \(\widehat{\overline{{\mathbf {Y}}}}^{ECGREG}=\widehat{\overline{{\mathbf {Y}}}}^{PHT}+{\widehat{N}}^{-1}\left( \sum \limits _{i\in {s_{B}}}d_{i}^{B}x_{i}^{\text {T}}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}x_{i}^{\text {T}} \right) \left( \sum \limits _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}x_{i}x_{i}^{\text {T}} \right) ^{-1}\sum \limits _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}x_{i}y_{i}\) is the population mean estimator based on estimated control generalized regression model, and \(\widehat{\overline{{\mathbf {Y}}}}^{PHT}={\widehat{N}}^{-1}\left( {\mathbf {d}}^{A} \right) ^{\text {T}}{\mathbf {y}}\). \(\square\)
Proof of Lemma 2
The estimated control model-assisted calibrated weights can be expressed as \(w_{i}=d_{i}^{A}g_{i}\), where \(g_{i}=1/\left( 1-q_{i}^{A}\nu _{i}\lambda \right)\) by the standard Lagrange multiplier method. The Lagrange multiplier \(\lambda =\left( \lambda _{1},\lambda _{2} \right) ^{\text {T}}\) can be obtained by solving the following equation
$$\begin{aligned} \sum \limits _{i\in {s_{A}}}\frac{d_{i}^{A}\nu _{i}}{1-q_{i}^{A}\nu _{i}\lambda }-\sum \limits _{i\in {s_{B}}}d_{i}^{B}\nu _{i}=0. \end{aligned}$$
(A7)
Define \(d_{i}^{A}\nu _{i}=d_{i}^{A}\nu _{i}\left( 1-q_{i}^{A}\nu _{i}\lambda +q_{i}^{A}\nu _{i}\lambda \right)\), and substitute it into (A7), we can obtain
$$\begin{aligned}&\sum \limits _{i\in {s_{A}}}\frac{d_{i}^{A}\nu _{i}\left( 1-q_{i}^{A}\nu _{i}\lambda +q_{i}^{A}\nu _{i}\lambda \right) }{1-q_{i}^{A}\nu _{i}\lambda }-\sum \limits _{i\in {s_{B}}}d_{i}^{B}\nu _{i}=0,\nonumber \\&\sum \limits _{i\in {s_{A}}}d_{i}^{A}\nu _{i}+\sum \limits _{i\in {s_{A}}}\frac{\lambda ^{\text {T}}d_{i}^{A}q_{i}^{A}\nu _{i}^{\text {T}}\nu _{i}}{1-q_{i}^{A}\nu _{i}\lambda }-\sum \limits _{i\in {s_{B}}}d_{i}^{B}\nu _{i}=0,\nonumber \\&\lambda ^{\text {T}}\sum \limits _{i\in {s_{A}}}\frac{d_{i}^{A}q_{i}^{A}\nu _{i}^{\text {T}}\nu _{i}}{1-\lambda ^{\text {T}}q_{i}^{A}\nu _{i}^{\text {T}}}=\sum \limits _{i\in {s_{B}}}d_{i}^{B}\nu _{i}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}\nu _{i}. \end{aligned}$$
(A8)
It follows from (A8) that
$$\begin{aligned} \frac{\lambda ^{*\text {T}}}{1+\lambda ^{*\text {T}}u^{*}}\sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}\nu _{i}^{\text {T}}\nu _{i}\le {{\mathbf {K}}}, \end{aligned}$$
(A9)
where \(\lambda ^{*}=(\left| \lambda _{1}\right| , \left| \lambda _{2}\right| )^{\text {T}}\), \(u^{*}=\left( \max \limits _{i\in {s_{A}}}\left| q_{i}^{A}\right| ,\max \limits _{i\in {s_{A}}}\left| q_{i}^{A}{\hat{\mu }}_{i}\right| \right) ^{\text {T}}\) and its order is \(o_{p}\left( n_{A}^{1/2} \right)\) by the condition (B1), and \({\mathbf {K}}=\left( \left| \sum \nolimits _{i\in {s_{B}}}d_{i}^{B}-\sum \nolimits _{i\in {s_{A}}}d_{i}^{A}\right| ,\left| \sum \nolimits _{i\in {s_{B}}}d_{i}^{B}{\hat{\mu }}_{i}-\sum \nolimits _{i\in {s_{A}}}d_{i}^{A}{\hat{\mu }}_{i}\right| \right)\). The condition (B2) implies \(\widehat{{\mathbf {T}}}^{M}_{A}={\mathbf {T}}^{M}+O_{p}\left( Nn_{A}^{-1/2} \right)\), where \(\widehat{{\mathbf {T}}}^{M}_{A}=\left( \widehat{{\mathbf {T}}}^{M}_{A1},\widehat{{\mathbf {T}}}^{M}_{A2} \right) =\left( \sum \limits _{i\in {s_{A}}}d_{i}^{A},\sum \limits _{i\in {s_{A}}}d_{i}^{A}{\hat{\mu }}_{i} \right)\), and the condition (B3) implies \(\widehat{{\mathbf {T}}}^{M}_{B}={\mathbf {T}}^{M}+O_{p}\left( Nn_{B}^{-1/2} \right)\). Furthermore, we can obtain \(\widehat{{\mathbf {T}}}^{{\mathbf {M}}}_{B}-\widehat{{\mathbf {T}}}^{{\mathbf {M}}}_{A}=O_{p}\left( Nn_{*}^{-1/2} \right)\), where \(n_{*}=\min \left( n_{A},n_{B} \right)\). Under conditions (B1)–(B4), we have \(\sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}\nu _{i}^{\text {T}}\nu _{i}=O\left( N \right)\), \(\lambda ^{\text {T}}=O_{p}\left( n_{*}^{-1/2} \right)\) and \(\max \limits _{i\in {s_{A}}}|\lambda ^{\text {T}}q_{i}^{A}\nu _{i}|=o_{p}(1)\). Furthermore, we can obtain
$$\begin{aligned}&\left( \lambda ^{ECMCGPEL} \right) ^{\text {T}}\nonumber \\&\quad =\left( \sum \limits _{i\in {s_{B}}}d_{i}^{B}\nu _{i}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}\nu _{i} \right) \left( \sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}\nu _{i}^{\text {T}}\nu _{i} \right) ^{-1}\left( 1+o_{p}\left( 1 \right) \right) \nonumber \\&\quad =\left( \sum \limits _{i\in {s_{B}}}d_{i}^{B}\nu _{i}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}\nu _{i} \right) \left( \sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}\nu _{i}^{\text {T}}\nu _{i} \right) ^{-1} \nonumber \\&\qquad +\left( \sum \limits _{i\in {s_{B}}}d_{i}^{B}\nu _{i}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}\nu _{i} \right) \left( \sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}\nu _{i}^{\text {T}}\nu _{i} \right) ^{-1}o_{p}\left( 1 \right) \nonumber \\&\quad =\left( \sum \limits _{i\in {s_{B}}}d_{i}^{B}\nu _{i}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}\nu _{i} \right) \left( \sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}\nu _{i}^{\text {T}}\nu _{i} \right) ^{-1}\nonumber \\&\qquad +O_{p}\left( Nn_{*}^{-1/2} \right) O_{p}\left( 1/N \right) o_{p}\left( 1 \right) \nonumber \\&\quad =\left( \sum \limits _{i\in {s_{B}}}d_{i}^{B}\nu _{i}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}\nu _{i} \right) \left( \sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}\nu _{i}^{\text {T}}\nu _{i} \right) ^{-1}+o_{p}\left( n_{*}^{-1/2} \right) . \nonumber \end{aligned}$$
Then, we can obtain
$$\begin{aligned} \lambda ^{ECMCGPEL}&=\left( \sum \limits _{i\in {s_{A}}}d_{i}^{A}q_{i}^{A}\nu _{i}^{\text {T}}\nu _{i} \right) ^{-1}\left[ \left( {\mathbf {d}}^{B} \right) ^{\text {T}}{\mathbf {M}}_{B}-\left( {\mathbf {d}}^{A} \right) ^{\text {T}}{\mathbf {M}}_{A} \right] ^{\text {T}}+o_{p}\left( n_{*}^{-1/2} \right) \nonumber \\&=\left( {\mathbf {M}}_{A}^{\text {T}}{\mathbf {D}}^{A}{\mathbf {Q}}^{A}{\mathbf {M}}_{A} \right) ^{-1}\left[ \left( {\mathbf {d}}^{B} \right) ^{\text {T}}{\mathbf {M}}_{B}-\left( {\mathbf {d}}^{A} \right) ^{\text {T}}{\mathbf {M}}_{A} \right] ^{\text {T}}+o_{p}\left( n_{*}^{-1/2} \right) , \end{aligned}$$
(A10)
where \({\mathbf {d}}^{B}=\left( d_{1}^{B},d_{2}^{B},\ldots ,d_{n_{B}}^{B} \right) ^{\text {T}}\), \({\mathbf {M}}_{B}=({\mathbf {M}}_{B1}, {\mathbf {M}}_{B2})=\left( {\mathbf {1}}_{(n_{B} \times 1)},\left( {\hat{\mu }}_{i} \right) _{i\in {s_{B}}} \right)\). By Taylor’s expansion, we can obtain \(g_{i}=1/\left( 1-q_{i}^{A}x_{i}^{\text {T}}\lambda \right) =1+q_{i}^{A}x_{i}^{\text {T}}\lambda +o_{p}\left( n_{A}^{-1/2} \right) \doteq 1+q_{i}^{A}x_{i}^{\text {T}}\lambda\). Then, the estimator of estimated control model-assisted calibrated weights is given by
$$\begin{aligned}&{\mathbf {W}}^{ECMCGPEL}\nonumber \\&\quad ={\mathbf {D}}^{A}\left[ {\mathbf {E}}+{\mathbf {Q}}^{A}{\mathbf {M}}_{A}\lambda ^{ECMCGPEL}+o_{p}\left( n_{A}^{-1/2} \right) \right] \nonumber \\&\quad ={\mathbf {d}}^{A}+{\mathbf {D}}^{A}{\mathbf {Q}}^{A}{\mathbf {M}}_{A}\left( {\mathbf {M}}_{A}^{\text {T}}{\mathbf {D}}^{A}{\mathbf {Q}}^{A}{\mathbf {M}}_{A} \right) ^{-1}\left[ \left( {\mathbf {d}}^{B} \right) ^{\text {T}}{\mathbf {M}}_{B}-\left( {\mathbf {d}}^{A} \right) ^{\text {T}}{\mathbf {M}}_{A} \right] ^{\text {T}}+o_{p}\left( n_{*}^{-1/2} \right) \nonumber \\&\qquad +o_{p}\left( n_{A}^{-1/2} \right) \nonumber \\&\quad ={\mathbf {d}}^{A}+{\mathbf {D}}^{A}{\mathbf {Q}}^{A}{\mathbf {M}}_{A}\left( {\mathbf {M}}_{A}^{\text {T}}{\mathbf {D}}^{A}{\mathbf {Q}}^{A}{\mathbf {M}}_{A} \right) ^{-1}\left[ \left( {\mathbf {d}}^{B} \right) ^{\text {T}}{\mathbf {M}}_{B}-\left( {\mathbf {d}}^{A} \right) ^{\text {T}}{\mathbf {M}}_{A} \right] ^{\text {T}}+o_{p}\left( n_{*}^{-1/2} \right) . \end{aligned}$$
Thus, the ECMCGPEL estimator of the population mean is
$$\begin{aligned}&\widehat{\overline{{\mathbf {Y}}}}^{ECMCGPEL}\nonumber \\&\quad ={\widehat{N}}^{-1}\left( {\mathbf {W}}^{ECMCGPEL} \right) ^{\text {T}}{\mathbf {y}}\nonumber \\&\quad ={\widehat{N}}^{-1}\left( {\mathbf {d}}^{A} \right) ^{\text {T}}{\mathbf {y}}+{\widehat{N}}^{-1}\left[ \left( {\mathbf {d}}^{B} \right) ^{\text {T}}{\mathbf {M}}_{B}-\left( {\mathbf {d}}^{A} \right) ^{\text {T}}{\mathbf {M}}_{A} \right] \left( {\mathbf {M}}_{A}^{\text {T}}{\mathbf {Q}}^{A}{\mathbf {D}}^{A}{\mathbf {M}}_{A} \right) ^{-1}\nonumber
\\&\qquad \cdot {\mathbf {M}}_{A}^{\text {T}}{\mathbf {Q}}^{A}{\mathbf {D}}^{A}{\mathbf {y}}+o_{p}\left( n_{*}^{-1/2} \right) \nonumber \\&\quad ={\widehat{N}}^{-1}\sum \limits _{i\in {s_{A}}}d_{i}^{A}y_{i}+{\widehat{N}}^{-1}\left( \sum _{i\in {s_{B}}}d_{i}^{B}{\hat{\mu }}_{i}-\sum _{i\in {s_{A}}}d_{i}^{A}{\hat{\mu }}_{i} \right) \widehat{{\mathbf {H}}}^{MC}+o_{p}\left( n_{*}^{-1/2} \right) , \end{aligned}$$
(A11)
where
$$\begin{aligned} \widehat{{\mathbf {H}}}^{MC}= & {} \frac{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}({\hat{\mu }}_{i}-\hat{{\bar{\mu }}})\left( y_{i}-{\bar{y}} \right) }{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}({\hat{\mu }}_{i}-\hat{{\bar{\mu }}})^{2}},\\ \hat{{\bar{\mu }}}= & {} \frac{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}{\hat{\mu }}_{i}}{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}},\\ {\bar{y}}= & {} \frac{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}y_{i}}{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}}. \end{aligned}$$
\(\square\)
Proof of Lemma 3
By taking the variance of equation in (3.6), we have
$$\begin{aligned}&V\Big (\widehat{\overline{{\mathbf {Y}}}}^{ECMCGPEL}\Big )\nonumber \\&\quad \doteq V_{{\mathcal {A}}}\Big ({\widehat{N}}^{-1}\sum \limits _{i\in {s_{A}}}d_{i}^{A}y_{i}+{\widehat{N}}^{-1}\Big (\sum \limits _{i\in {s_{B}}}d_{i}^{B}{\hat{\mu }}_{i}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}{\hat{\mu }}_{i}\Big )\widehat{{\mathbf {H}}}^{MC}\Big ) \nonumber \\&\quad = {\widehat{N}}^{-2}V_{{\mathcal {A}}}\Big [\sum \limits _{i\in {s_{A}}}d_{i}^{A}(y_{i} - {\hat{\mu }}_{i}\widehat{{\mathbf {H}}}^{MC}) + \sum \limits _{i\in {s_{B}}}d_{i}^{B}{\hat{\mu }}_{i}\widehat{{\mathbf {H}}}^{MC}\Big ] \nonumber \\&\quad ={\widehat{N}}^{-2}\bigg \{V_{{\mathcal {A}}}\Big [E_{{\mathcal {B}}}\Big (\sum \limits _{i\in {s_{A}}}d_{i}^{A}(y_{i} - {\hat{\mu }}_{i}\widehat{{\mathbf {H}}}^{MC}) + \sum \limits _{i\in {s_{B}}}d_{i}^{B}{\hat{\mu }}_{i}\widehat{{\mathbf {H}}}^{MC}\Big )\Big ] \nonumber \\&\qquad + E_{{\mathcal {A}}}\Big [V_{{\mathcal {B}}}\Big (\sum \limits _{i\in {s_{A}}}d_{i}^{A}(y_{i} - {\hat{\mu }}_{i}\widehat{{\mathbf {H}}}^{MC}) + \sum \limits _{i\in {s_{B}}}d_{i}^{B}{\hat{\mu }}_{i}\widehat{{\mathbf {H}}}^{MC}\Big )\Big ]\bigg \}\nonumber \\&\quad = {\widehat{N}}^{-2}\bigg \{V_{{\mathcal {A}}}\Big [\sum \limits _{i\in {s_{A}}}d_{i}^{A}(y_{i} - {\hat{\mu }}_{i}\widehat{{\mathbf {H}}}^{MC}) + \sum \limits _{i\in {U}}{\hat{\mu }}_{i}\widehat{{\mathbf {H}}}^{MC}\Big ] + V_{{\mathcal {B}}}\Big (\sum \limits _{i\in {s_{B}}}d_{i}^{B}{\hat{\mu }}_{i}\widehat{{\mathbf {H}}}^{MC}\Big )\bigg \}\nonumber \\&\quad = {\widehat{N}}^{-2} \bigg \{\sum _{i\in {U}}\bigg (\frac{y_{i}-{\hat{\mu }}_{i}\widehat{{\mathbf {H}}}^{MC}}{\pi _{i}^{A}}\bigg )^{2}\pi _{i}^{A}(1-\pi _{i}^{A}) + \sum _{i\in {U}}\bigg (\frac{{\hat{\mu }}_{i}\widehat{{\mathbf {H}}}^{MC}}{\pi _{i}^{B}}\bigg )^{2}\pi _{i}^{B}(1-\pi _{i}^{B})\nonumber \\&\qquad + \sum _{i\in {U}}\sum _{j \ne i} (\pi _{ij}^{A}-\pi _{i}^{A}\pi _{j}^{A}) \frac{y_{i}-{\hat{\mu }}_{i}\widehat{{\mathbf {H}}}^{MC}}{\pi _{i}^{A}} \frac{y_{j}-{\hat{\mu }}_{j}\widehat{{\mathbf {H}}}^{MC}}{\pi _{j}^{A}}\nonumber \\&\qquad + \sum _{i\in {U}}\sum _{j \ne i} (\pi _{ij}^{B}-\pi _{i}^{B}\pi _{j}^{B}) \frac{{\hat{\mu }}_{i}\widehat{{\mathbf {H}}}^{MC}}{\pi _{i}^{B}} \frac{{\hat{\mu }}_{j}\widehat{{\mathbf {H}}}^{MC}}{\pi _{j}^{B}}\bigg \}\,. \end{aligned}$$
(A12)
Thus, Lemma 3 holds. \(\square\)
Proof of Theorem 1
Under the condition (C6), the SCAD regression satisfies the oracle property according to Theorems 1 and 2 in Fan and Li (2001), that is
$$\begin{aligned}&\text {Pr}\left( \widehat{{\mathbf {B}}}^{\left( 2 \right) }=0 \right) \rightarrow 1,\nonumber \\ \sqrt{n_{A}}\left\{ I_{1}\left( {\mathbf {B}}^{\left( 1 \right) } \right) +\Sigma \right\}&\left\{ \widehat{{\mathbf {B}}}^{\left( 1 \right) }-{\mathbf {B}}^{\left( 1 \right) }+\left( I_{1}\left( {\mathbf {B}}^{\left( 1 \right) } \right) +\Sigma \right) ^{-1}{\mathbf {b}}\right\} \rightarrow {N}\left\{ 0,I_{1}\left( {\mathbf {B}}^{\left( 1 \right) } \right) \right\} , \end{aligned}$$
where \(I_{1}\left( {\mathbf {B}}^{\left( 1 \right) } \right) =I_{1}\left( {\mathbf {B}}^{\left( 1 \right) },0 \right)\) is the Fisher information of \({\mathbf {B}}^{\left( 1 \right) }\) under known \({\mathbf {B}}^{\left( 2 \right) }=0\), \(\Sigma =\text {diag}\left\{ p''_{\lambda _{n_{A}}}\left( \left| {\mathbf {B}}_{1}\right| \right) ,\ldots ,p''_{\lambda _{n_{A}}}\left( \left| {\mathbf {B}}_{q}\right| \right) \right\}\), \({\mathbf {b}}=\left( p'_{\lambda _{n_{A}}}\left( \left| {\mathbf {B}}_{1}\right| \right) \text {sgn}\left( {\mathbf {B}}_{1} \right) ,\ldots ,p'_{\lambda _{n_{A}}}\left( \left| {\mathbf {B}}_{1}\right| \right) \text {sgn}\left( {\mathbf {B}}_{q} \right) \right)\), and \({\mathbf {B}}^{\left( 1 \right) }=\left( {\mathbf {B}}_{1},\ldots ,{\mathbf {B}}_{q} \right) ^{\text {T}}\). If \(\max \left\{ \left| p''_{\lambda _{n_{A}}}\left( \left| {\mathbf {B}}_{j}\right| \right) \right| :{\mathbf {B}}_{j}\ne 0,j=1,2,\ldots ,q\right\} \rightarrow 0\), we have \(\widehat{{\mathbf {B}}}^{\left( 1 \right) }={\mathbf {B}}+O_{p}\left( n_{A}^{-1/2} \right)\).
We use SCAD to estimate the model parameter \(\beta\) based on the non-probability sample \(s_{A}\). We can then obtain the SCAD coefficient estimator \(\widehat{{\mathbf {B}}}\) of \(\beta\). Similar to the proof of Lemma 2, the ECMC-SCAD-GPEL estimator of the population mean can be given by
$$\begin{aligned}&\widehat{\overline{{\mathbf {Y}}}}^{ECMCSCADGPEL}\nonumber \\&\quad ={\widehat{N}}^{-1}\left( {\mathbf {W}}^{ECMCSCADGPEL} \right) ^{\text {T}}{\mathbf {y}}\nonumber \\&\quad ={\widehat{N}}^{-1}\left( {\mathbf {d}}^{A} \right) ^{\text {T}}{\mathbf {y}}+{\widehat{N}}^{-1}\left[ \left( {\mathbf {d}}^{B} \right) ^{\text {T}}{\mathbf {M}}_{B}-\left( {\mathbf {d}}^{A} \right) ^{\text {T}}{\mathbf {M}}_{A} \right] \left( {\mathbf {M}}_{A}^{\text {T}}{\mathbf {Q}}^{A}{\mathbf {D}}^{A}{\mathbf {M}}_{A} \right) ^{-1}\nonumber \\&\qquad \cdot {\mathbf {M}}_{A}^{\text {T}}{\mathbf {Q}}^{A}{\mathbf {D}}^{A}{\mathbf {y}}+o_{p}\left( n_{*}^{-1/2} \right) \nonumber \\&\quad ={\widehat{N}}^{-1}\sum \limits _{i\in {s_{A}}}d_{i}^{A}y_{i}+{\widehat{N}}^{-1}\left( \sum _{i\in {s_{B}}}d_{i}^{B}{\hat{\mu }}_{i}-\sum _{i\in {s_{A}}}d_{i}^{A}{\hat{\mu }}_{i} \right) \widehat{{\mathbf {H}}}^{MC}+o_{p}\left( n_{*}^{-1/2} \right) , \end{aligned}$$
(A13)
where
$$\begin{aligned} \widehat{{\mathbf {H}}}^{MC}= & {} \frac{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}({\hat{\mu }}_{i}-\hat{{\bar{\mu }}})\left( y_{i}-{\bar{y}} \right) }{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}({\hat{\mu }}_{i}-\hat{{\bar{\mu }}})^{2}},\\ \hat{{\bar{\mu }}}= & {} \frac{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}{\hat{\mu }}_{i}}{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}},\\ {\bar{y}}= & {} \frac{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}y_{i}}{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}}. \end{aligned}$$
Now we derive the consistency of population mean estimator \(\widehat{\overline{{\mathbf {Y}}}}^{ECMCSCADGPEL}\). Under conditions (C1)-(C3), the second order Taylor series expansion of \(\mu (x_{i},\widehat{{\mathbf {B}}})\) around \({\mathbf {B}}\) is:
$$\begin{aligned}&\mu (x_{i},\widehat{{\mathbf {B}}})=\mu (x_{i},{\mathbf {B}})+\left\{ \frac{\mu \left( x_{i},s \right) }{\partial {s}}|_{s={\mathbf {B}}}\right\} ^{\text {T}}\left( \widehat{{\mathbf {B}}}-{\mathbf {B}} \right) \nonumber \\&\quad + \left( \widehat{{\mathbf {B}}}-{\mathbf {B}} \right) ^{\text {T}}\left\{ \frac{\partial ^{2}{\mu \left( x_{i},s \right) }}{\partial {s}\partial {s}^{\text {T}}}|_{s=\mathbf {B^{*}}}\right\} \left( \widehat{{\mathbf {B}}}-{\mathbf {B}} \right) , \end{aligned}$$
(A14)
for \({\mathbf {B}}^{*}\in (\widehat{{\mathbf {B}}},{\mathbf {B}})\) or \(({\mathbf {B}},\widehat{{\mathbf {B}}})\). Let
$$\begin{aligned} {\mathbf {f}}\left( x_{i},{\mathbf {B}} \right)&=\frac{\partial \mu \left( x_{i},s \right) }{\partial {s}}\mid _{s={\mathbf {B}}},\nonumber \\ {\mathbf {g}}\left( x_{i},{\mathbf {B}}^{*} \right)&=\frac{\partial ^{2}{\mu \left( x_{i},s \right) }}{\partial {s}\partial {s}^{\text {T}}}|_{s=\mathbf {B^{*}}}. \end{aligned}$$
Note that \({\mathbf {f}}\) is a vector of the length p and \({\mathbf {g}}\) is a matrix of size \(p\times {p}\), where p is the number of parameters \(\beta\). Under conditions (C2) and (C3)
$$\begin{aligned} \max _{i}\left| {\mathbf {f}}\left( x_{i},{\mathbf {B}} \right) \right|&\le {f\left( x_{i},{\mathbf {B}} \right) },\nonumber \\ \max _{k,j}\left| {\mathbf {g}}\left( x_{i},{\mathbf {B}}^{*} \right) \right|&\le {g\left( x_{i},{\mathbf {B}}^{*} \right) }. \end{aligned}$$
(A15)
By Condition (C1), we use the probability sample \(s_{B}\) to estimate the population mean based on the second order Taylor series expansion of \(\mu (x_{i},\widehat{{\mathbf {B}}})\) around \({\mathbf {B}}\), that is
$$\begin{aligned}&N^{-1}\sum \limits _{i\in {s_{B}}}d_{i}^{B}\mu (x_{i},\widehat{{\mathbf {B}}})\nonumber \\&\quad =N^{-1}\sum \limits _{i\in {s_{B}}}d_{i}^{B}\mu (x_{i},{\mathbf {B}})+N^{-1}\sum \limits _{i\in {s_{A}}}d_{i}^{B}{\mathbf {f}}^{\text {T}}(x_{i},{\mathbf {B}})\left( \widehat{{\mathbf {B}}}-{\mathbf {B}} \right) \nonumber \\&\qquad +N^{-1}\sum \limits _{i\in {s_{B}}}d_{i}^{B}\left( \widehat{{\mathbf {B}}}-{\mathbf {B}} \right) ^{\text {T}}{\mathbf {g}}(x_{i},\mathbf {B^{*}})\left( \widehat{{\mathbf {B}}}-{\mathbf {B}} \right) \nonumber \\&\quad =N^{-1}\sum \limits _{i\in {s_{B}}}d_{i}^{B}\mu (x_{i},{\mathbf {B}})+O_{p}\left( n_{B}^{-1/2} \right) +O_{p}\left( n_{B}^{-1/2} \right) O_{p}\left( n_{B}^{-1/2} \right) \nonumber \\&\quad =N^{-1}\sum \limits _{i\in {s_{B}}}d_{i}^{B}\mu (x_{i},{\mathbf {B}})+O_{p}\left( n_{B}^{-1/2} \right) . \end{aligned}$$
(A16)
By the Eq. (A16), we can obtain \(\sum \limits _{i\in {s_{B}}}d_{i}^{B}\mu (x_{i},\widehat{{\mathbf {B}}})=\sum \limits _{i\in {s_{B}}}d_{i}^{B}\mu (x_{i},{\mathbf {B}})+O_{p}\left( Nn_{B}^{-1/2} \right)\). Combing this result and the condition (C5), we have
$$\begin{aligned} \frac{\sum _{i\in {s_{B}}}d_{i}^{B}\mu (x_{i},\widehat{{\mathbf {B}}})}{{\widehat{N}}}&=\frac{\sum _{i\in {s_{B}}}d_{i}^{B}\mu (x_{i},{\mathbf {B}})+O_{p}\left( Nn_{B}^{-1/2} \right) }{{\widehat{N}}}\nonumber \\&=\frac{\sum _{i\in {s_{B}}}d_{i}^{B}\mu (x_{i},{\mathbf {B}})+O_{p}\left( Nn_{B}^{-1/2} \right) }{N+O_{p}\left( Nn_{B}^{-1/2} \right) }\nonumber \\&=N^{-1}\sum _{i\in {s_{B}}}d_{i}^{B}\mu (x_{i},{\mathbf {B}})+o_{p}\left( 1 \right) . \end{aligned}$$
(A17)
Similarly, we use the non-probability sample \(s_{A}\) to estimate the population mean. Based on the second order Taylor series expansion of \(\mu (x_{i},\widehat{{\mathbf {B}}})\) around \({\mathbf {B}}\), we have
$$\begin{aligned}&N^{-1}\sum \limits _{i\in {s_{A}}}d_{i}^{A}\mu (x_{i},\widehat{{\mathbf {B}}})\nonumber \\&\quad =N^{-1}\sum \limits _{i\in {s_{A}}}d_{i}^{A}\mu (x_{i},{\mathbf {B}})+N^{-1}\sum \limits _{i\in {s_{A}}}d_{i}^{A}{\mathbf {f}}^{\text {T}}(x_{i},{\mathbf {B}})\left( \widehat{{\mathbf {B}}}-{\mathbf {B}} \right) \nonumber \\&\qquad +N^{-1}\sum \limits _{i\in {s_{A}}}d_{i}^{A}\left( \widehat{{\mathbf {B}}}-{\mathbf {B}} \right) ^{\text {T}}{\mathbf {g}}(x_{i},\mathbf {B^{*}})\left( \widehat{{\mathbf {B}}}-{\mathbf {B}} \right) \nonumber \\&\quad =N^{-1}\sum \limits _{i\in {s_{A}}}d_{i}^{A}\mu (x_{i},{\mathbf {B}})+O_{p}\left( n_{A}^{-1/2} \right) +O_{p}\left( n_{A}^{-1/2} \right) O_{p}\left( n_{A}^{-1/2} \right) \nonumber \\&\quad =N^{-1}\sum \limits _{i\in {s_{A}}}d_{i}^{A}\mu (x_{i},{\mathbf {B}})+O_{p}\left( n_{A}^{-1/2} \right) . \end{aligned}$$
(A18)
It follows that \(\sum \limits _{i\in {s_{A}}}d_{i}^{A}\mu (x_{i},\widehat{{\mathbf {B}}})=\sum \limits _{i\in {s_{A}}}d_{i}^{A}\mu (x_{i},{\mathbf {B}})+O_{p}\left( Nn_{A}^{-1/2} \right)\). By condition (C5), we can further get
$$\begin{aligned} \frac{\sum _{i\in {s_{A}}}d_{i}^{A}\mu (x_{i},\widehat{{\mathbf {B}}})}{{\widehat{N}}}&=\frac{\sum _{i\in {s_{A}}}d_{i}^{A}\mu (x_{i},{\mathbf {B}})+O_{p}\left( Nn_{A}^{-1/2} \right) }{{\widehat{N}}}\nonumber \\&=\frac{\sum _{i\in {s_{A}}}d_{i}^{A}\mu (x_{i},{\mathbf {B}})+O_{p}\left( Nn_{A}^{-1/2}
\right) }{N+O_{p}\left( Nn_{A}^{-1/2} \right) }\nonumber \\&=N^{-1}\sum _{i\in {s_{A}}}d_{i}^{A}\mu (x_{i},{\mathbf {B}})+o_{p}\left( 1 \right) . \end{aligned}$$
(A19)
By conditions (C6), (A17) and (A19), we can obtain
$$\begin{aligned}&{\widehat{N}}^{-1}\sum _{i\in {s_{B}}}d_{i}^{B}\mu (x_{i},\widehat{{\mathbf {B}}})-{\widehat{N}}^{-1}\sum _{i\in {s_{A}}}d_{i}^{A}\mu (x_{i},\widehat{{\mathbf {B}}})\nonumber \\&\quad =N^{-1}\sum _{i\in {s_{B}}}d_{i}^{B}\mu (x_{i},{\mathbf {B}})-N^{-1}\sum _{i\in {s_{A}}}d_{i}^{A}\mu (x_{i},{\mathbf {B}}) +o_{p}\left( 1 \right) -o_{p}\left( 1 \right) \nonumber \\&\quad =N^{-1}\sum _{i\in {s_{B}}}d_{i}^{B}\mu (x_{i},{\mathbf {B}})-N^{-1}\sum _{i\in {s_{A}}}d_{i}^{A}\mu (x_{i},{\mathbf {B}}) +o_{p}\left( 1 \right) . \end{aligned}$$
(A20)
Besides, we also have
$$\begin{aligned} \hat{{\bar{\mu }}}=&\sum \limits _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}\mu (x_{i},\widehat{{\mathbf {B}}})/\sum \limits _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}\nonumber \\ =&\left( \sum \limits _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A} \right) ^{-1}\bigg (\sum \limits _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}\Big [\mu (x_{i},{\mathbf {B}}) +{\mathbf {f}}^{\text {T}}(x_{i},{\mathbf {B}})(\widehat{{\mathbf {B}}}-{\mathbf {B}})\nonumber \\&+(\widehat{{\mathbf {B}}}-{\mathbf {B}})^{\text {T}}{\mathbf {g}}(x_{i},\mathbf {B^{*}})(\widehat{{\mathbf {B}}}-{\mathbf {B}})\Big ]\bigg )\nonumber \\ =&\left( \sum \limits _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A} \right) ^{-1}\nonumber \\&\left( \sum \limits _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}\left[ \mu (x_{i},{\mathbf {B}})+O_{p}\left( n_{A}^{-1/2} \right) +O_{p}\left( n_{A}^{-1/2} \right) O_{p}\left( n_{A}^{-1/2} \right) \right] \right) \nonumber \\ =&\sum \limits _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}\mu (x_{i},{\mathbf {B}})/\sum \limits _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}+O_{p}\left( n_{A}^{-1/2} \right) \nonumber \\ =&{\bar{\mu }}+O_{p}\left( n_{A}^{-1/2} \right) . \end{aligned}$$
(A21)
where \({\bar{\mu }}=\sum \limits _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}\mu (x_{i},{\mathbf {B}})/\sum \limits _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}=\sum \limits _{i\in {U}}q_{i}\mu (x_{i},{\mathbf {B}})/\sum \limits _{i\in {U}}q_{i}\). By Eqs. (A14) and (A21), we have
$$\begin{aligned}&N^{-1}\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}({\hat{\mu }}_{i}-\hat{{\bar{\mu }}})\nonumber \\&\quad =N^{-1}\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}\left[ \mu (x_{i},{\mathbf {B}})+{\mathbf {f}}^{\text {T}}(x_{i},{\mathbf {B}})(\widehat{{\mathbf {B}}}-{\mathbf {B}}) +(\widehat{{\mathbf {B}}}-{\mathbf {B}})^{\text {T}}{\mathbf {g}}(x_{i},\mathbf {B^{*}})(\widehat{{\mathbf {B}}}-{\mathbf {B}})-\hat{{\bar{\mu }}} \right] \nonumber \\&\quad =N^{-1}\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}\bigg [\mu (x_{i},{\mathbf {B}})+{\mathbf {f}}^{\text {T}}(x_{i},{\mathbf {B}})(\widehat{{\mathbf {B}}}-{\mathbf {B}}) +(\widehat{{\mathbf {B}}}-{\mathbf {B}})^{\text {T}}{\mathbf {g}}(x_{i},\mathbf {B^{*}})(\widehat{{\mathbf {B}}}-{\mathbf {B}})\nonumber \\&\qquad -{\bar{\mu }}-O_{p}\left( n_{A}^{-1/2} \right) \bigg ]\nonumber \\&\quad =N^{-1}\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}(\mu _{i}-{\bar{\mu }})+O_{p}\left( n_{A}^{-1/2} \right) +O_{p}\left( n_{A}^{-1} \right) -O_{p}\left( n_{A}^{-1/2} \right) \nonumber \\&\quad =N^{-1}\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}(\mu _{i}-{\bar{\mu }})+O_{p}\left( n_{A}^{-1/2} \right) . \end{aligned}$$
(A22)
According to (A22), we have
$$\begin{aligned} N^{-1}\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}({\hat{\mu }}_{i}-\hat{{\bar{\mu }}})^{2} = N^{-1}\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}(\mu _{i}-{\bar{\mu }})^{2}+O_{p}\left( n_{A}^{-1} \right) \,. \end{aligned}$$
(A23)
Under Eqs. (A22) and (A23), we have
$$\begin{aligned} \widehat{{\mathbf {H}}}^{MC}&=\frac{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}({\hat{\mu }}_{i}-\hat{{\bar{\mu }}})\left( y_{i}-{\bar{y}} \right) }{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}({\hat{\mu }}_{i}-\hat{{\bar{\mu }}})^{2}} =\frac{N^{-1}\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}({\hat{\mu }}_{i}-\hat{{\bar{\mu }}})\left( y_{i}-{\bar{y}} \right) }{N^{-1}\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}({\hat{\mu }}_{i}-\hat{{\bar{\mu }}})^{2}}\nonumber \\&=\frac{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}(\mu _{i}-{\bar{\mu }})\left( y_{i}-{\bar{y}} \right) +O_{p}\left( n_{A}^{-1/2} \right) }{\sum _{i\in {s_{A}}}q_{i}^{A}d_{i}^{A}(\mu _{i}-{\bar{\mu }})^{2}+O_{p}\left( n_{A}^{-1} \right) }\nonumber \\&\rightarrow \quad {{\mathbf {H}}}^{MC} \quad as \quad n_{A}\rightarrow \infty . \end{aligned}$$
(A24)
Therefore, we can obtain \(\widehat{{\mathbf {H}}}^{MC}={\mathbf {H}}^{MC}+o_{p}\left( 1 \right)\). In addition, we also have
$$\begin{aligned} \frac{\sum _{i\in {s_{A}}}d_{i}^{A}y_{i}}{{\widehat{N}}}&=\frac{\sum _{i\in {s_{A}}}d_{i}^{A}y_{i}}{N+O_{p}\left( Nn_{A}^{-1/2} \right) }\nonumber \\&=\frac{\sum _{i\in {s_{A}}}d_{i}^{A}y_{i}}{N}+O_{p}\left( n_{A}^{-1/2} \right) . \end{aligned}$$
(A25)
Furthermore, by (A13), (A20), (A24) and (A25), we can obtain
$$\begin{aligned}&\widehat{\overline{{\mathbf {Y}}}}^{ECMCSCADGPEL}\nonumber \\&\quad = {\widehat{N}}^{-1}\sum \limits _{i\in {s_{A}}}d_{i}^{A}y_{i} +\left[ N^{-1}\sum _{i\in {s_{B}}}d_{i}^{B}\mu \left( x_{i},{\mathbf {B}} \right) -N^{-1}\sum _{i\in {s_{A}}}d_{i}^{A}\mu \left( x_{i},{\mathbf {B}} \right) +o_{p}\left( 1 \right) \right] \nonumber \\&\qquad \cdot \left( {\mathbf {H}}^{MC}+o_{p}\left( 1 \right) \right) +o_{p}\left( n_{*}^{-1/2} \right) \nonumber \\&\quad = N^{-1}\sum \limits _{i\in {s_{A}}}d_{i}^{A}y_{i}+O_{p}\left( n_{A}^{-1/2} \right) +N^{-1}\left[ \sum _{i\in {s_{B}}}d_{i}^{B}\mu \left( x_{i},{\mathbf {B}} \right) -\sum _{i\in {s_{A}}}d_{i}^{A}\mu \left( x_{i},{\mathbf {B}} \right) \right] {\mathbf {H}}^{MC}\nonumber \\&\qquad +o_{p}\left( n_{*}^{-1/2} \right) \nonumber \\&\quad = N^{-1}\sum \limits _{i\in {s_{A}}}d_{i}^{A}y_{i} +N^{-1}\left[ \sum _{i\in {s_{B}}}d_{i}^{B}\mu \left( x_{i},{\mathbf {B}} \right) -\sum _{i\in {s_{A}}}d_{i}^{A}\mu \left( x_{i},{\mathbf {B}} \right) \right] {\mathbf {H}}^{MC}+O_{p}\left( n_{A}^{-1/2} \right) , \end{aligned}$$
(A26)
where
$$\begin{aligned} {\mathbf {H}}^{MC}=\frac{\sum _{i\in {U}}q_{i}(\mu _{i}-{\bar{\mu }})\left( y_{i}-{\bar{y}} \right) }{\sum _{i\in {U}}q_{i}(\mu _{i}-{\bar{\mu }})^{2}}, \quad {\bar{\mu }}=\frac{\sum _{i\in {U}}q_{i}{\mu }_{i}}{\sum _{i\in {U}}q_{i}}, \quad {\bar{y}}=\frac{\sum _{i\in {U}}q_{i}{y}_{i}}{\sum _{i\in {U}}q_{i}}. \end{aligned}$$
Therefore, Theorem 1 holds. \(\square\)
Proof of Theorem 2
$$\begin{aligned} \text {E}_{\xi }\left[ {\mathbf {H}}^{MC} \right]&=\text {E}_{\xi }\left[ \frac{\sum _{i\in {U}}q_{i}\left( \mu _{i}-{\bar{\mu }} \right) \left( y_{i}-{\bar{y}} \right) }{\sum _{i\in {U}}q_{i}\left( \mu _{i}-{\bar{\mu }} \right) ^{2}} \right] =\frac{\sum _{i\in {U}}q_{i}\left( \mu _{i}-{\bar{\mu }} \right) \left( \mu _{i}-{\bar{\mu }} \right) }{\sum _{i\in {U}}q_{i}\left( \mu _{i}-{\bar{\mu }} \right) ^{2}}=1\,. \end{aligned}$$
Combining \(\text {E}_{\xi }\left[ y_{i} \right] =\mu _{i}\) and \(\text {E}_{\xi }\left[ {\mathbf {H}}^{MC} \right] =1\), we can get
$$\begin{aligned}&\text {E}_{{\mathcal {B}}}\left[ \text {E}_{\xi }\bigg (\widehat{\overline{{\mathbf {Y}}}}^{ECMCSCADGPEL}-\overline{{\mathbf {Y}}}\bigg ) \right] \nonumber \\&\quad \dot{=} \text {E}_{{\mathcal {B}}}\left[ \text {E}_{\xi }\bigg (N^{-1}\sum _{i\in {s_{A}}}d_{i}^{A} (y_{i} - \mu _{i}{\mathbf {H}}^{MC}) +N^{-1}\sum _{i\in {s_{B}}}d_{i}^{B}\mu _{i}-N^{-1}\sum _{i\in {U}}y_{i}\bigg ) \right] \nonumber \\&\quad = N^{-1}\text {E}_{{\mathcal {B}}}\bigg \{\text {E}_{\xi }\Big [\sum _{i\in {s_{A}}}d_{i}^{A} (y_{i} - \mu _{i}{\mathbf {H}}^{MC}) +\sum _{i\in {s_{B}}}d_{i}^{B}\mu _{i}-\sum _{i\in {U}}y_{i}\Big ]\bigg \}\nonumber \\&\quad = N^{-1} \text {E}_{{\mathcal {B}}}\left[ \sum _{i\in {s_{A}}}d_{i}^{A}(\mu _{i}-\mu _{i}) + \sum _{i\in {s_{B}}}d_{i}^{B}\mu _{i}-\sum _{i\in {U}}\mu _{i} \right] \nonumber \\&\quad = N^{-1}\sum _{i\in {U}}\mu _{i}-N^{-1}\sum _{i\in {U}}\mu _{i}\nonumber \\&\quad =0\,. \end{aligned}$$
(A27)
Thus, Theorem 2 holds. \(\square\)
Proof of Theorem 3
By taking the variance of equation in (4.5), we have
$$\begin{aligned}&V\Big (\widehat{\overline{{\mathbf {Y}}}}^{ECMCSCADGPEL}\Big )\nonumber \\&\quad \doteq V_{{\mathcal {A}}}\Big (N^{-1}\sum \limits _{i\in {s_{A}}}d_{i}^{A}y_{i}+N^{-1}\Big (\sum \limits _{i\in {s_{B}}}d_{i}^{B}\mu _{i}-\sum \limits _{i\in {s_{A}}}d_{i}^{A}\mu _{i}\Big ){\mathbf {H}}^{MC}\Big ) \nonumber \\&\quad = N^{-2}V_{{\mathcal {A}}}\Big [\sum \limits _{i\in {s_{A}}}d_{i}^{A}(y_{i} - \mu _{i}{\mathbf {H}}^{MC}) + \sum \limits _{i\in {s_{B}}}d_{i}^{B}\mu _{i}{\mathbf {H}}^{MC}\Big ] \nonumber \\&\quad = N^{-2}\bigg \{V_{{\mathcal {A}}}\Big [E_{{\mathcal {B}}}\Big (\sum \limits _{i\in {s_{A}}}d_{i}^{A}(y_{i} - \mu _{i}{\mathbf {H}}^{MC}) + \sum \limits _{i\in {s_{B}}}d_{i}^{B}\mu _{i}{\mathbf {H}}^{MC}\Big )\Big ] \nonumber \\&\qquad + E_{{\mathcal {A}}}\Big [V_{{\mathcal {B}}}\Big (\sum \limits _{i\in {s_{A}}}d_{i}^{A}(y_{i} - \mu _{i}{\mathbf {H}}^{MC}) + \sum \limits _{i\in {s_{B}}}d_{i}^{B}\mu _{i}{\mathbf {H}}^{MC}\Big )\Big ]\bigg \}\nonumber \\&\quad = N^{-2}\bigg \{V_{{\mathcal {A}}}\Big [\sum \limits _{i\in {s_{A}}}d_{i}^{A}(y_{i} - \mu _{i}{\mathbf {H}}^{MC}) + \sum \limits _{i\in {U}}\mu _{i}{\mathbf {H}}^{MC}\Big ] + V_{{\mathcal {B}}}\Big (\sum \limits _{i\in {s_{B}}}d_{i}^{B}\mu _{i}{\mathbf {H}}^{MC}\Big )\bigg \}\nonumber \\&\quad = N^{-2} \bigg \{\sum _{i\in {U}}\bigg (\frac{y_{i}-\mu _{i}{\mathbf {H}}^{MC}}{\pi _{i}^{A}}\bigg )^{2}\pi _{i}^{A}(1-\pi _{i}^{A}) + \sum _{i\in {U}}\bigg (\frac{\mu _{i}{\mathbf {H}}^{MC}}{\pi _{i}^{B}}\bigg )^{2}\pi _{i}^{B}(1-\pi _{i}^{B})\nonumber \\&\qquad + \sum _{i\in {U}}\sum _{j \ne i} (\pi _{ij}^{A}-\pi _{i}^{A}\pi _{j}^{A}) \frac{y_{i}-\mu _{i}{\mathbf {H}}^{MC}}{\pi _{i}^{A}} \frac{y_{j}-\mu _{j}{\mathbf {H}}^{MC}}{\pi _{j}^{A}}\nonumber \\&\qquad + \sum _{i\in {U}}\sum _{j \ne i} (\pi _{ij}^{B}-\pi _{i}^{B}\pi _{j}^{B}) \frac{\mu _{i}{\mathbf {H}}^{MC}}{\pi _{i}^{B}} \frac{\mu _{j}{\mathbf {H}}^{MC}}{\pi _{j}^{B}}\bigg \}\,. \end{aligned}$$
(A28)
Thus, Theorem 3 holds. \(\square\)
