4. Composite estimation for matrix sampling design (d)

Takis Merkouris

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4.1 Core set of variables with known totals

We discuss first a special case of the matrix sampling design (d) in which the variables that are common to the three samples have known totals. In this very realistic sampling setting, all samples collect also information on the same vector of auxiliary variables $z$ for which the vector of population totals $t_z$ is known. For illustration we consider again three samples, as in Figure 2.1 (but with $z$ added in all subsamples). Then, the CGR estimator ${\widehat{X}}^{\text{CGR}}$ in (3.1) may be augmented with the ordinary regression terms ${\widehat{B}}_{3x}\left(t_z-{\widehat{Z}}_1\right)+{\widehat{B}}_{4x}\left(t_z-{\widehat{Z}}_2\right)+{\widehat{B}}_{5x}\left(t_z-{\widehat{Z}}_3\right),$ where ${\widehat{Z}}_i\mathrm{,}i=\mathrm{1,2,3}$ is the HT estimator of $t_z$ based on sample $S_i;$ similarly for ${\widehat{Y}}^{\text{CGR}}.$ This estimator has improved efficiency, as it incorporates additional information, and is generated by a calibration procedure that includes the additional three constraints ${\widehat{Z}}_i^{\text{CGR}}=t_z,$ and has the design matrix $\mathcal{X}$ in (2.7) augmented with the block-diagonal matrix $Z=\text{diag}\left\{Z_1\mathrm{,}{Z}_2\mathrm{,}{Z}_3\right\}.$ In the simplest case when the sample matrices $Z_i$ reduce to the unit columns $1_i$ (with corresponding total the size of the population), the calibration scheme is the one specified in Corollary 1 above. As shown in the proof of the next theorem, an application of Lemma 1 to the present calibration procedure, with partitioned design matrix $\left(\mathcal{X}\mathrm{,}Z\right),R=\Lambda$ and calibration totals ${\left(0^{\prime }\mathrm{,}{0}^{\prime }\mathrm{,}{t^{\prime }}_z\mathrm{,}{t^{\prime }}_z\mathrm{,}{t^{\prime }}_z\right)}^{\prime },$ gives a modified CGR form of (3.1) with GR estimators incorporating information on $z$ in place of HT estimators. This is compactly written as ${\widehat{\mathcal{X}}}_3^{\text{GR}}-\widehat{ℬ}{\widehat{\mathcal{X}}}^{\text{GR}},$ where ${\widehat{\mathcal{X}}}_3^{\text{GR}}={\widehat{\mathcal{X}}}_3+{{\mathcal{X}}^{\prime }}_3\Lambda Z{\left(Z^{\prime }\Lambda Z\right)}^{-1}\left(t_{(z)}-\widehat{Z}\right),$ with $t_{(z)}={\left({t^{\prime }}_z\mathrm{,}{t^{\prime }}_z\mathrm{,}{t^{\prime }}_z\right)}^{\prime },$ and ${\widehat{\mathcal{X}}}^{\text{GR}}$ expressed similarly, and where $\widehat{ℬ}=\left[{{\mathcal{X}}^{\prime }}_3\Lambda \left(I-P_Z\right)\mathcal{X}\right]{\left[{\mathcal{X}}^{\prime }\Lambda \left(I-P_Z\right)\mathcal{X}\right]}^{-1}$ with $P_Z=Z{\left(Z^{\prime }\Lambda Z\right)}^{-1}{Z}^{\prime }\Lambda .$

Replacing $\Lambda$ by ${\Lambda }^0$ in the calibration procedure gives the optimal composite regression estimator, compactly written as ${\widehat{\mathcal{X}}}_3^{\text{OR}}-{\widehat{ℬ}}^o{\widehat{\mathcal{X}}}^{\text{OR}},$ with optimal regression estimators incorporating information on $z$ in place of GR estimators, and with ${\widehat{ℬ}}^o=\left[{{\mathcal{X}}^{\prime }}_3{\Lambda }^0\left(I-P_Z^0\right)\mathcal{X}\right]{\left[{\mathcal{X}}^{\prime }{\Lambda }^0\left(I-P_Z^0\right)\mathcal{X}\right]}^{-1}$ where $P_Z^0=Z{\left(Z^{\prime }{\Lambda }^{0}Z\right)}^{-1}{Z}^{\prime }{\Lambda }^0.$ Noticing that $\left(I-P_Z^0\right){\mathcal{X}}_3$ is the matrix of residuals corresponding to ${\widehat{\mathcal{X}}}_3^{\text{OR}}$ and that ${{\mathcal{X}}^{\prime }}_3{\Lambda }^0\left(I-P_Z^0\right)\mathcal{X}={{\mathcal{X}}^{\prime }}_3{\left(I-P_Z^0\right)}^{\prime }{\Lambda }^0\left(I-P_Z^0\right)\mathcal{X}=\widehat{\text{AC}}\left({\widehat{\mathcal{X}}}_3^{\text{OR}}\mathrm{,}{\widehat{\mathcal{X}}}^{\text{OR}}\right),$ and similarly for $\widehat{\text{AV}}\left({\widehat{\mathcal{X}}}^{\text{OR}}\right),$ it follows that

$${\widehat{ℬ}}^o=-\widehat{\text{AC}}\left[\begin{array}{c}\left(\begin{array}{c}{\widehat{X}}_3^{\text{OR}}\\ {\widehat{Y}}_3^{\text{OR}}\end{array}\right)\mathrm{,}\left(\begin{array}{c}{\widehat{X}}_1^{\text{OR}}-{\widehat{X}}_3^{\text{OR}}\\ {\widehat{Y}}_2^{\text{OR}}-{\widehat{Y}}_3^{\text{OR}}\end{array}\right)\end{array}\right]{\left[\widehat{\text{AV}}\left(\begin{array}{l}{\widehat{X}}_1^{\text{OR}}-{\widehat{X}}_3^{\text{OR}}\hfill \\ {\widehat{Y}}_2^{\text{OR}}-{\widehat{Y}}_3^{\text{OR}}\hfill \end{array}\right)\right]}^{-1}\mathrm{,}(4.1)$$

in analogy with (2.4), or with (2.5) in non-nested sampling. Thus, ${\widehat{ℬ}}^o$ is optimal in the sense of minimizing the approximate variance of the estimator ${\widehat{\mathcal{X}}}_3^{\text{OR}}-{\widehat{ℬ}}^o{\widehat{\mathcal{X}}}^{\text{OR}},$ which is then asymptotically BLUE. An alternative estimator, of weaker optimality, has the form ${\widehat{\mathcal{X}}}_3^{\text{GR}}-{\widehat{ℬ}}^{wo}{\widehat{\mathcal{X}}}^{\text{GR}},$ where the coefficient ${\widehat{ℬ}}^{wo}=\left[{{\mathcal{X}}^{\prime }}_3{\left(I-P_Z\right)}^{\prime }{\Lambda }^0\left(I-P_Z\right)\mathcal{X}\right]{\left[{\mathcal{X}}^{\prime }{\left(I-P_Z\right)}^{\prime }{\Lambda }^0\left(I-P_Z\right)\mathcal{X}\right]}^{-1}$ has the form (4.1) but with GR estimators in place of OR estimators. This estimator, differing from the CGR only in the regression coefficient, is optimal in the restricted sense of being the composite of GR estimators incorporating information on $z$ that has minimum approximate variance. In general, this later composite estimator cannot be obtained as a calibration estimator. The following theorem gives conditions under which the CGR estimator is optimal in one of the two senses in non-nested matrix sampling; the proof is given in the Appendix. The nested sampling version of the theorem, with subsampling schemes and proof as in Theorem 1, is omitted for brevity.

Theorem 2 Consider the following sampling strategies.

• $\left(a\right)$ For all three samples $S_1,S_2$ and $S_3$ assume SRS with sampling fractions $f_i=n_i/N,$ and specify all constants $q_{ik}$ in ${\Lambda }_i$ as $q_{ik}=\left(n_i-1\right)/N\left(1-f_i\right).$ Consider the augmented design matrix $\mathcal{Z}=\left(\mathcal{X}\mathrm{,}Z\right)$ in (2.7), where $Z=\text{diag}\left\{Z_1\mathrm{,}{Z}_2\mathrm{,}{Z}_3\right\},$ and with the corresponding augmented vector of calibration totals $t_{\mathcal{Z}}={\left(0^{\prime }\mathrm{,}{0}^{\prime }\mathrm{,}{t^{\prime }}_z\mathrm{,}{t^{\prime }}_z\mathrm{,}{t^{\prime }}_z\right)}^{\prime }.$ Further, suppose that $Z_i{h}_i=1$ for constant vectors  $h_i.$

• Then, the calibration procedure gives the CGR as ${\widehat{\mathcal{X}}}_3^{\text{GR}}-\widehat{ℬ}{\widehat{\mathcal{X}}}^{\text{GR}}={\widehat{\mathcal{X}}}_3^{\text{GR}}-{\widehat{ℬ}}^{wo}{\widehat{\mathcal{X}}}^{\text{GR}},$ i.e., the CGR estimator is the optimal composite of GR estimators incorporating information on $z.$

• $\left(b\right)$   For all three samples $S_1,S_2$ and $S_3$ assume STRSRS with sampling fraction $f_{ih}=n_{ih}/N_{ih}$ in stratum $h$ of sample $i,h=1,\dots \mathrm{,}{H}_i$ and $N_{ih}$ denoting stratum size, and specify the constants in ${\Lambda }_i$ as $q_{ik}=\left(n_{ih}-1\right)/N_h\left(1-f_{ih}\right)$ for all units of stratum $h.$ Further, assume that within each sample the units are sorted by stratum, and consider the augmented design matrix $\mathcal{Z}=\left(\mathcal{X}\mathrm{,}Z\mathrm{,}D\right)$ in (2.7), with corresponding augmented vector of calibration totals $t_{\mathcal{Z}}={(0^{\prime }\mathrm{,}{0}^{\prime }\mathrm{,}{t^{\prime }}_z\mathrm{,}{t^{\prime }}_z,{t^{\prime }}_z\mathrm{,}{N^{\prime }}_1\mathrm{,}{N^{\prime }}_2\mathrm{,}{N^{\prime }}_3)}^{\prime }.$ The definition of $D$ and $N_i$ is as before.

• Then, the calibration procedure gives the CGR as ${\widehat{\mathcal{X}}}_3^{\text{OR}}-{\widehat{ℬ}}^o{\widehat{\mathcal{X}}}^{\text{OR}},$ i.e., the CGR estimator is the optimal composite of optimal regression estimators incorporating information on $z.$

• $\left(c\right)$   For all three samples $S_1,S_2$ and $S_3$ assume stratified Poisson sampling and specify the constants $q_{ik}$ in the entries of ${\Lambda }_i$ as $q_{ik}={\pi }_{ihk}/\left(1-{\pi }_{ihk}\right)$ for the units of stratum $h.$

• Then, the calibration procedure, with $\mathcal{Z}$ and $t_{\mathcal{Z}}$ as in $\left(a\right),$  gives the CGR as ${\widehat{\mathcal{X}}}_3^{\text{GR}}-\widehat{ℬ}{\widehat{\mathcal{X}}}^{\text{GR}}={\widehat{\mathcal{X}}}_3^{\text{OR}}-{\widehat{ℬ}}^o{\widehat{\mathcal{X}}}^{\text{OR}},$ i.e., GR and OR estimators are identical, and the CGR estimator is the optimal composite of optimal regression estimators incorporating information on  $z.$

The condition $Z_i{h}_i=1$ in $\left(a\right)$ of Theorem 2 is customarily satisfied when the vector $z$ contains categorical variables. Results analogous to Corollaries 1 and 2 of the previous section hold also for parts $\left(b\right)$ and $\left(c\right)$ of Theorem 2. Here too, for sampling designs other than those assumed in Theorem 2, the value $q_{ik}={\tilde{n}}_i/\left({\tilde{n}}_1+{\tilde{n}}_2+{\tilde{n}}_3\right)$ in the entries of $\Lambda$ should be used.

Finally, by analogy to (3.2), and with the appropriate decomposition of the vector of calibrated weights $c,$ the composite estimator ${\widehat{X}}^{\text{CGR}}$ takes now the form

$${\widehat{X}}^{\text{CGR}}={\widehat{B}}_{1x}{\widehat{X}}_1^{\text{GR}}+\left(I-{\widehat{B}}_{1x}\right){\widehat{X}}_3^{\text{GR}}\mathrm{,}$$

where ${\widehat{X}}_1^{\text{GR}}$ and ${\widehat{X}}_3^{\text{GR}}$ are GR estimators using information on $z$ from $S_1,$ and on $y$ and $z$ from $S_2$ and $S_3,$ respectively, and ${\widehat{B}}_{1x}$ is the corresponding matrix regression coefficient. Similar is the expression for ${\widehat{Y}}^{\text{CGR}}.$ Of course, ${\widehat{X}}^{\text{CGR}}$ and ${\widehat{Y}}^{\text{CGR}}$ can be obtained directly through this modified $c$ in the simple linear forms ${\widehat{X}}^{\text{CGR}}={X^{\prime }}_3{c}_3$ and ${\widehat{Y}}^{\text{CGR}}={Y^{\prime }}_3{c}_3.$

4.2 Core set of variables with unknown totals

We turn now to the case of matrix sampling design (d) in which the variables $z$ that are common to the three samples have unknown totals. Estimation in this setting includes the construction of a composite estimator of the vector of totals $t_z.$ In line with the formulation of Section 2, composite estimators of $t_x,t_y$ and $t_z$ that are best linear unbiased combinations of the HT estimators ${\widehat{X}}_1,{\widehat{Z}}_1,{\widehat{Y}}_2,{\widehat{Z}}_2,{\widehat{X}}_3,{\widehat{Y}}_3,{\widehat{Z}}_3$ are given by

         $$\begin{array}{lll}{\widehat{X}}^B\hfill & =\hfill & B_{1x}{\widehat{X}}_1+\left(I-B_{1x}\right){\widehat{X}}_3+B_{3x}\left({\widehat{Y}}_2-{\widehat{Y}}_3\right)+B_{2x}\left({\widehat{Z}}_1-{\widehat{Z}}_3\right)+B_{4x}\left({\widehat{Z}}_2-{\widehat{Z}}_3\right)\hfill \\ {\widehat{Y}}^B\hfill & =\hfill & B_{3y}{\widehat{Y}}_2+\left(I-B_{3y}\right){\widehat{Y}}_3+B_{1y}\left({\widehat{X}}_1-{\widehat{X}}_3\right)+B_{2y}\left({\widehat{Z}}_1-{\widehat{Z}}_3\right)+B_{4y}\left({\widehat{Z}}_2-{\widehat{Z}}_3\right)\hfill \\ {\widehat{Z}}^B\hfill & =\hfill & B_{2z}{\widehat{Z}}_1+B_{4z}{\widehat{Z}}_2+\left(I-B_{2z}-B_{4z}\right){\widehat{Z}}_3+B_{1z}\left({\widehat{X}}_1-{\widehat{X}}_3\right)+B_{3z}\left({\widehat{Y}}_2-{\widehat{Y}}_3\right).\hfill \end{array}(4.2)$$

The estimators in (4.2) can be written in the matrix regression form

$$\left(\begin{array}{c}{\widehat{X}}^B\\ {\widehat{Y}}^B\\ {\widehat{Z}}^B\end{array}\right)=\left(\begin{array}{c}{\widehat{X}}_3\\ {\widehat{Y}}_3\\ {\widehat{Z}}_3\end{array}\right)+ℬ\left(\begin{array}{c}{\widehat{X}}_1-{\widehat{X}}_3\\ {\widehat{Z}}_1-{\widehat{Z}}_3\\ {\widehat{Y}}_2-{\widehat{Y}}_3\\ {\widehat{Z}}_2-{\widehat{Z}}_3\end{array}\right)\mathrm{,}(4.3)$$

with the variance-minimizing matrix of coefficients given by $ℬ=-\text{Cov}\left(u_3\mathrm{,}{u}_{12}-u_3^{\star }\right){\left[V\left(u_{12}-u_3^{\star }\right)\right]}^{-1},$ where $u_3={\left({{\widehat{X}}^{\prime }}_3\mathrm{,}{{\widehat{Y}}^{\prime }}_3\mathrm{,}{{\widehat{Z}}^{\prime }}_3\right)}^{\prime },$ $u_3^{\star }={({{\widehat{X}}^{\prime }}_3\mathrm{,}{{\widehat{Z}}^{\prime }}_3\mathrm{,}{{\widehat{Y}}^{\prime }}_3\mathrm{,}{{\widehat{Z}}^{\prime }}_3)}^{\prime },$ $u_{12}={\left({{\widehat{X}}^{\prime }}_1\mathrm{,}{{\widehat{Z}}^{\prime }}_1\mathrm{,}{{\widehat{Y}}^{\prime }}_2\mathrm{,}{{\widehat{Z}}^{\prime }}_2\right)}^{\prime }.$ With estimated covariance and variance matrices we obtain the estimated optimal matrix ${\widehat{ℬ}}^o,$ and (4.3) becomes then an optimal multivariate regression estimator. Then, proceeding as in Section 2, it can be shown that

$${\widehat{ℬ}}^o=({{\mathcal{X}}^{\prime }}_{3-}{\Lambda }^0\mathcal{X}){({\mathcal{X}}^{\prime }{\Lambda }^0\mathcal{X})}^{-1}\mathrm{,}$$

where

$$\mathcal{X}=\left(\begin{array}{cccc}-X_1& -Z_1& 0& 0\\ 0& 0& -Y_2& -Z_2\\ X_3& Z_3& Y_3& Z_3\end{array}\right)(4.4)$$

is the design matrix corresponding to the regression estimator (4.3), ${\mathcal{X}}_{3-}$ is the matrix $\mathcal{X}$ with the second column eliminated and the first two rows set equal to zero, and ${\Lambda }^0$ is as in Section 2.

Replacing the matrix ${\Lambda }^0$ with the weighting matrix $\Lambda ,$ gives the generalized regression coefficient $\widehat{ℬ}=({{\mathcal{X}}^{\prime }}_{3-}\Lambda \mathcal{X}){({\mathcal{X}}^{\prime }\Lambda \mathcal{X})}^{-1},$ and (4.3) becomes the CGR estimator of ${\left({t^{\prime }}_x\mathrm{,}{t^{\prime }}_y\mathrm{,}{t^{\prime }}_z\right)}^{\prime }$

$$\left(\begin{array}{c}{\widehat{X}}^{\text{CGR}}\\ {\widehat{Y}}^{\text{CGR}}\\ {\widehat{Z}}^{\text{CGR}}\end{array}\right)=\left(\begin{array}{c}{\widehat{X}}_3\\ {\widehat{Y}}_3\\ {\widehat{Z}}_3\end{array}\right)+\widehat{ℬ}\left(\begin{array}{c}{\widehat{X}}_1-{\widehat{X}}_3\\ {\widehat{Z}}_1-{\widehat{Z}}_3\\ {\widehat{Y}}_2-{\widehat{Y}}_3\\ {\widehat{Z}}_2-{\widehat{Z}}_3\end{array}\right)\mathrm{.}(4.5)$$

The estimator (4.5) can be conveniently obtained through a calibration procedure that gives a vector of calibrated weights for the combined sample $S$ having the form $c=w+\Lambda \mathcal{X}{\left({\mathcal{X}}^{\prime }\Lambda \mathcal{X}\right)}^{-1}\left(0-{\mathcal{X}}^{\prime }w\right),$ as before, but now satisfying the additional constraint ${\widehat{Z}}_1^{\text{CGR}}={\widehat{Z}}_2^{\text{CGR}}={\widehat{Z}}_3^{\text{CGR}}.$ Expression (4.5) is then obtained simply as ${{\mathcal{X}}^{\prime }}_{3-}c,$ based on sample $S_3.$

The explicit expression (4.2), different for the optimal regression and the generalized regression variants only in the form of the linear coefficients, shows that the composite estimators of $t_x$ and $t_y$ are more efficient than their counterparts in matrix sampling design (c), equation (2.2), because they incorporate information on the common variables $z,$ assuming non-zero correlation with $x$ and $y.$ Particularly remarkable is the expression for the composite estimator of $t_z:$ it involves a linear combination of the three HT estimators of $t_z$ derived from the three samples, plus the two regression terms implying additional efficiency through the correlation of $z$ with $x$ and $y.$ One would expect the additional terms to be zero because an optimal combination of the three estimators should incorporate all information on $z$ available in the three samples. In general, however, the associated coefficients are not zero. In non-nested sampling, conditions under which these coefficients are zero are given by the following proposition, the proof of which is given in the Appendix. The result should also hold in nested sampling.

Proposition 1 The coefficients $B_{1z}$ and $B_{3z}$ in the estimator ${\widehat{Z}}^B$ in (4.2) are zero only if

$$\begin{array}{lll}{\left[V\left({\widehat{Z}}_1\right)\right]}^{-1}\text{Cov}\left({\widehat{X}}_1\mathrm{,}{\widehat{Z}}_1\right)\hfill & =\hfill & {\left[V\left({\widehat{Z}}_3\right)\right]}^{-1}\text{Cov}\left({\widehat{X}}_3\mathrm{,}{\widehat{Z}}_3\right)\hfill \\ {\left[V\left({\widehat{Z}}_2\right)\right]}^{-1}\text{Cov}\left({\widehat{Y}}_2\mathrm{,}{\widehat{Z}}_2\right)\hfill & =\hfill & {\left[V\left({\widehat{Z}}_3\right)\right]}^{-1}\text{Cov}\left({\widehat{Y}}_3\mathrm{,}{\widehat{Z}}_3\right)\mathrm{.}\hfill \end{array}(4.6)$$

This can happen only if the sampling designs for the three samples are identical, including equal sample sizes, or only if the sampling design across samples is the same design with equal inclusion probability for all units, but not necessarily with the same sample size.

Noticing that the quantities on each side of the equations (4.6) are regression coefficients, according to Proposition 1 the terms of the estimator ${\widehat{Z}}^B$ incorporating the correlation of $z$ with $x$ and $y$ are zero only if the effect of the regression of $x$ and $y$ on $z$ is identical in samples $S_1$ and $S_3$ and in samples $S_2$ and $S_3,$ respectively. The essence of this finding is that estimation of $t_z$ using only information on $z$ from the three samples, but ignoring information on $x$ and $y,$ will be suboptimal when there is differential regression effect of $x$ and $y$ on $z$ in the various samples. The efficiency of ${\widehat{Z}}^B$ relative to the composite estimator ${\tilde{Z}}^B$ that uses only information on $z$ was possible to gauge in the simple setting involving scalar $x,y$ and $z,$ simple random sampling for $S_1$ and $S_3$ and Bernoulli sampling for $S_2,$ and equal sampling rates for all three samples. Then only the first equation of (4.6) holds. After much tedious algebra the efficiency of ${\widehat{Z}}^B$ relative to ${\tilde{Z}}^B$ was derived to be $\left[V\left({\tilde{Z}}^B\right)-V\left({\widehat{Z}}^B\right)/V\left({\tilde{Z}}^B\right)\right]=G/H,$ with

$$\begin{array}{lll}G\hfill & =\hfill & 2\left(r_{xz}^2-1\right){\left(r_{yz}c{v}_y-c{v}_z\right)}^2\hfill \\ H\hfill & =\hfill & \left(c{v}_z^2+1\right)\left(\left(12-9r_{yz}^2\right)r_{xz}^2-3r_{xy}\left(2r_{yz}{r}_{xz}-1\right)+12\left(r_{yz}^2-1\right)\right)c{v}_z^{2}c{v}_y^2\hfill \\ \hfill & +\hfill & 2\left(r_{xy}^2+r_{yz}^2\right)c{v}_y^2+8\left(r_{xz}^2-1\right)c{v}_y^2-4r_{yz}{r}_{xy}{r}_{xz}c{v}_y^2\hfill \\ \hfill & +\hfill & 6\left(r_{xz}^2-1\right)c{v}_z\left(c{v}_z-2r_{yz}c{v}_y\right)\hfill \end{array}$$

where $r_{xy},r_{xz}$ and $r_{yz}$ denote population correlation coefficients, and $c{v}_y,c{v}_z$ denote coefficients of variation. Although in this setting the departure from the conditions of Proposition 1 is minimal, different configurations of admissible values for $r_{xy},r_{xz},r_{yz},c{v}_y$ and $c{v}_z$ show that the efficiency gain may be substantial, making up for the inefficiency of the HT estimator of $t_z$ based on the Bernoulli sample $S_2.$ For example, when $r_{xy}=0.3,r_{xz}=0.3,r_{yz}=0.3$ and $c{v}_y=0.1,c{v}_z=0.6,$ the efficiency gain is 23%. In the case of the composite optimal regression estimator ${\widehat{Z}}^{\text{COR}},$ with estimated coefficients ${\widehat{B}}_{1z}^o$ and ${\widehat{B}}_{3z}^o,$ the regression coefficients in (4.6) are estimated, and thus the equalities in (4.6) would never hold exactly because of the sample differences. Likewise in the case of the CGR estimator ${\widehat{Z}}^{\text{CGR}},$ for which equations formally identical to (4.6) are given in terms of sample generalized regression coefficients.

Regarding the efficiency of the CGR estimator (4.5), an exact analogue of Theorem 1 holds in the present setting, with the same sampling strategies for which the CGR estimator is optimal regression estimator and asymptotically BLUE.

Composite estimation for a matrix sampling scheme involving a core set of variables with both known and unknown totals can be carried out using the obvious extended calibration scheme.

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