Optimal linear estimation in two-phase sampling
Section 5. A two-phase generalized regression estimator

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A variant of $\widehat{\text{B}}={\stackrel{⌣}{\Psi }}^{\prime }\widehat{\Delta }\text{X}{\mathrm{<mo\; stretchy="false">(</mo>}{\text{X\hspace{0.17em}}}^{\prime }\widehat{\Delta }\text{X}\mathrm{<mo\; stretchy="false">)</mo>}}^{-1}$ that is computationally efficient, but generally suboptimal, is the generalized regression (GREG) coefficient ${\widehat{\text{B}}}^{\text{GR}}={\Psi }^{\prime }\Lambda \text{X}{\mathrm{<mo\; stretchy="false">(</mo>}{\text{X\hspace{0.17em}}}^{\prime }\Lambda \text{X}\mathrm{<mo\; stretchy="false">)</mo>}}^{-1},$ where $\Psi$ is as in (4.1) and with $y_k=0$ if $k\notin s_2,$ and $\Lambda$ is the “weighting” matrix $\text{diag}\mathrm{<mo\; stretchy="false">(</mo>}{\Lambda }_1\mathrm{,}{\Lambda }_2\mathrm{,}{\Lambda }_1\mathrm{<mo\; stretchy="false">)</mo>},$ with ${\Lambda }_1=\text{diag}\mathrm{<mo>\{</mo>}{w}_{1k}/q_{1k}\mathrm{<mo>\}</mo>}$ and ${\Lambda }_2=\text{diag}\mathrm{<mo>\{</mo>}{w}_k/q_{2k}\mathrm{<mo>\}</mo>},$ and with $q_{1k},$ $q_{2k}$ being positive constants. This gives the GREG estimator

$${\widehat{t}}_{\Psi }^{\text{GR}}={\widehat{t}}_{\Psi }+{\widehat{\text{B}}}^{\text{GR}}\mathrm{<mo\; stretchy="false">(</mo>}{t}_{\text{X}}-{\widehat{t}}_{\text{X}}\mathrm{<mo\; stretchy="false">)</mo>}={\widehat{\text{B}}}^{\text{GR}}{t}_{\text{X}}+\mathrm{<mo\; stretchy="false">(</mo>}\Psi -\text{X}{\widehat{\text{B}}}^{{\text{GR}}^{\prime }}{)}^{\prime }{w}^{*}\mathrm{.}(5.1)$$

Note that ${\widehat{\text{B}}}^{\text{GR}}$ is optimal in the sense of least squares, i.e., it minimizes the quadratic distance $\mathrm{<mo\; stretchy="false">(</mo>}\Psi -\text{X}{\widehat{\text{B}}}^{{\text{GR}}^{\prime }}{)}^{\prime }\Lambda \mathrm{<mo\; stretchy="false">(</mo>}\Psi -\text{X}{\widehat{\text{B}}}^{{\text{GR}}^{\prime }}),$ involving the residuals $\Psi -\text{X}{\widehat{\text{B}}}^{{\text{GR}}^{\prime }}$ in ${\widehat{t}}_{\Psi }^{\text{GR}},$ whereas the coefficient $\widehat{\text{B}}$ minimizes $\mathrm{<mo\; stretchy="false">(</mo>}\Psi -\text{X}\widehat{\text{B}}{)}^{\prime }\widehat{\Delta }\mathrm{<mo\; stretchy="false">(</mo>}\Psi -\text{X}\widehat{\text{B}}{)}^{\prime },$ the estimated approximate variance of the optimal estimator ${\widehat{t}}_{\Psi }^O.$ In this sense ${\widehat{t}}_{\Psi }^{\text{GR}}$ is an approximation to ${\widehat{t}}_{\Psi }^O.$ The two components of ${\widehat{t}}_{\Psi }^{\text{GR}},$ similar in structure to the components of ${\widehat{t}}_{\Psi }^O$ in (4.2), are

$$\begin{array}{ll}{\widehat{t}}_y^{\text{GR}}\hfill & ={\tilde{t}}_y+\left[Y_2^{\prime }{\Lambda }_2{X}_2+Y_1^{\prime }{\Lambda }_1{X}_1\right]{\left[X_2^{\prime }{\Lambda }_2{X}_2+X_1^{\prime }{\Lambda }_1{X}_1\right]}^{-1}\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{t}}_x-{\tilde{t}}_x\mathrm{<mo\; stretchy="false">)</mo>}\hfill \\ \hfill & +Y_1^{\prime }{\Lambda }_1{X}_{11}{\left[X_{11}^{\prime }{\Lambda }_1{X}_{11}\right]}^{-1}\mathrm{<mo\; stretchy="false">(</mo>}{t}_{x_1}-{\widehat{t}}_{x_1}\mathrm{<mo\; stretchy="false">)</mo>}\hfill \\ {\widehat{t}}_x^{\text{GR}}\hfill & ={\widehat{t}}_x+X_1^{\prime }{\Lambda }_1{X}_{11}{\left[X_{11}^{\prime }{\Lambda }_1{X}_{11}\right]}^{-1}\mathrm{<mo\; stretchy="false">(</mo>}{t}_{x_1}-{\widehat{t}}_{x_1}\mathrm{<mo\; stretchy="false">)</mo>.}\hfill \end{array}(5.2)$$

The GREG estimator ${\widehat{t}}_x^{\text{GR}}$ is the standard single-phase GREG estimator based on $s_1$ and the auxiliary variable $x_1.$ The GREG estimator ${\widehat{t}}_y^{\text{GR}},$ with the two orthogonal regression terms shown in (5.2), is expressed explicitly in terms of sample units as

$$\begin{array}{ll}{\widehat{t}}_y^{\text{GR}}\hfill & ={\tilde{t}}_y+\text{}\text{}\left[{\displaystyle {\sum }_{s_2}\mathrm{<mo\; stretchy="false">(</mo>}{\Lambda }_{1k}+{\Lambda }_{2k}\mathrm{<mo\; stretchy="false">)</mo>}{y}_k{x}_k^{\prime }}\right]{\left[{\displaystyle {\sum }_{s_2}{\Lambda }_{2k}{x}_k{x}_k^{\prime }}+{\displaystyle {\sum }_{s_1}{\Lambda }_{1k}{x}_k{x}_k^{\prime }}\right]}^{-1}\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{t}}_x-{\tilde{t}}_x\mathrm{<mo\; stretchy="false">)</mo>}\hfill \\ \hfill & +\left({\displaystyle {\sum }_{s_2}{\Lambda }_{1k}{y}_k{x}_{1k}^{\prime }}\right){\left({\displaystyle {\sum }_{s_1}{\Lambda }_{1k}{x}_{1k}{x}_{1k}^{\prime }}\right)}^{-1}\mathrm{<mo\; stretchy="false">(</mo>}{t}_{x_1}-{\widehat{t}}_{x_1}\mathrm{<mo\; stretchy="false">)</mo>,}\hfill \end{array}$$

where ${\Lambda }_{1k}=w_{1k}/q_{1k}$ and ${\Lambda }_{2k}=w_k/q_{2k}$ are the $k^{\text{th}}$ element of ${\Lambda }_1$ and ${\Lambda }_2,$ respectively. The constants $q_{ik}$ should be specified as $q_{ik}=n_i,$ to account for the differential in the sample size of $s_i;$ see Merkouris (2004) for a justification in the context of calibrating combined samples. An equivalent adjustment of the weights in ${\Lambda }_{1k}$ and ${\Lambda }_{2k}$ can be made through the multiplication of $w_{1k}$ in ${\Lambda }_{1k}$ by $\varphi =n_2/n_1.$ Values of $q_{ik}$ that convert the GREG estimator ${\widehat{t}}_y^{\text{GR}}$ to the optimal estimator ${\widehat{t}}_y^O$ can be specified for two-phase sampling designs for which optimal estimation is possible, as in the similar context of matrix sampling (Merkouris, 2015). For the simple example involving Poisson sampling in both phases, this specification is $q_{1k}={\pi }_{1k}/\mathrm{<mo\; stretchy="false">(</mo>\; 1}-{\pi }_{1k}\mathrm{<mo\; stretchy="false">)</mo>}$ and $q_{2k}={\pi }_{1k}{\pi }_{2k}/\mathrm{<mo\; stretchy="false">(</mo>\; 1}-{\pi }_{1k}{\pi }_{2k}\mathrm{<mo\; stretchy="false">)</mo>},$ rendering ${\Lambda }_1$ and ${\Lambda }_2$ identical to ${\widehat{\Delta }}_1$ and ${\widehat{\Delta }}_2.$

The vector of calibrated weights associated with the GREG estimator ${\widehat{t}}_{\Psi }^{\text{GR}}$ is $c^{\text{GR}}=w^{*}+\Lambda \text{X}{\mathrm{<mo\; stretchy="false">(</mo>}{\text{X\hspace{0.17em}}}^{\prime }\Lambda \text{X}\mathrm{<mo\; stretchy="false">)</mo>}}^{-1}(t_{\text{X}}-{\text{X\hspace{0.17em}}}^{\prime }{w}^{*}).$ It has the same form as $c^{*}$ in (4.3), but with ${\Lambda }_1$ and ${\Lambda }_2$ in place of $-{\Delta }_1$ and ${\Delta }_2,$ and minimizes the generalized least-squares distance $\mathrm{<mo\; stretchy="false">(</mo>}{c}^{\text{GR}}-w^{*}{)}^{\prime }{\Lambda }^{-1}\mathrm{<mo\; stretchy="false">(</mo>}{c}^{\text{GR}}-w^{*}\mathrm{<mo\; stretchy="false">)</mo>}$ subject to the constraints ${\text{X}}^{\prime }{c}^{\text{GR}}=t_{\text{X}}.$ The partition $\text{X}=\mathrm{<mo\; stretchy="false">(</mo>}{\text{X}}_{12}\mathrm{,}{\text{X}}_1\mathrm{<mo\; stretchy="false">)</mo>},$ defined after (4.3), allows the orthogonal decomposition of the vector $c^{*}$

$$\begin{array}{ll}{c}^{\text{GR}}\hfill & =w^{*}+\Lambda {\text{X}}_{12}{\mathrm{<mo\; stretchy="false">(</mo>}{\text{X\hspace{0.17em}}}_{12}^{\prime }\Lambda {\text{X\hspace{0.17em}}}_{12}\mathrm{<mo\; stretchy="false">)</mo>}}^{-1}(0-{\text{X\hspace{0.17em}}}_{12}^{\prime }{w}^{*})\hfill \\ \hfill & +\Lambda {\text{X}}_1{\mathrm{<mo\; stretchy="false">(</mo>}{\text{X\hspace{0.17em}}}_1^{\prime }\Lambda {\text{X\hspace{0.17em}}}_1\mathrm{<mo\; stretchy="false">)</mo>}}^{-1}(t_{x_1}-{\text{X\hspace{0.17em}}}_1^{\prime }{w}^{*})\mathrm{.}\hfill \end{array}(5.3)$$

In the right hand side of (5.3), the sum of the first and second terms would result from calibration with constraint ${\text{X\hspace{0.17em}}}_{12}^{\prime }{c}^{\text{GR}}=0$ only, while the sum of the first and third terms would result from calibration with constraint ${\text{X\hspace{0.17em}}}_1^{\prime }{c}^{\text{GR}}=t_{x_1}$ only. The practical implication of this is that the vector $c^{*}$ could be formed by concatenating the weight vectors generated by two separate calibrations, i.e., calibration of ${\mathrm{<mo\; stretchy="false">(</mo>}{w}_1^{\prime }\mathrm{,}{w}^{\prime }\mathrm{<mo\; stretchy="false">)</mo>}}^{\prime }$ using ${\mathrm{<mo\; stretchy="false">(</mo>}-X_1^{\prime }\mathrm{,}{X}_2^{\prime }\mathrm{<mo\; stretchy="false">)</mo>}}^{\prime }$ followed by calibration of $w_1$ using $X_{11}.$ However, the one-step calibration procedure generating $c^{\text{GR}}$ is more convenient.

On the basis of its Taylor linearization, the GREG estimator ${\widehat{t}}_y^{\text{GR}}$ in (5.1) is approximately (for large samples) unbiased. Furthermore, denoting by $e$ the matrix of sample residuals $\Psi -\text{X}{\widehat{\text{B}}}^{{\text{GR}}^{\prime }}$   , the estimated approximate variance of ${\widehat{t}}_y^{\text{GR}}={\widehat{\text{B}}}^{\text{GR}}{t}_{\text{X}}+e^{\prime }{w}^{*}$ is the estimated variance of $e^{\prime }{w}^{*},$ i.e., $\widehat{\text{AV}}\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{t}}_y^{\text{GR}}\mathrm{<mo\; stretchy="false">)</mo>}=\widehat{\text{Var}}\mathrm{<mo\; stretchy="false">(</mo>}{e}^{\prime }{w}^{*}\mathrm{<mo\; stretchy="false">)</mo>}=e^{\prime }\widehat{\Delta }e,$ whereas the estimated variance of the HT estimator ${\tilde{t}}_y$ is $\widehat{\text{Var}}({\tilde{t}}_y)=\widehat{\text{Var}}\mathrm{<mo\; stretchy="false">(</mo>}{\Psi }_1^{\prime }{w}^{\mathrm{<mo>*</mo>}}\mathrm{<mo\; stretchy="false">)</mo>}=Y_2^{\prime }{\widehat{\Delta }}_2{Y}_2,$ with ${\Psi }_1$ being the first column submatrix of $\Psi .$

Now using the calibration form ${\Psi }_1^{\prime }{c}^{\text{GR}}$ of ${\widehat{t}}_y^{\text{GR}}$ and the orthogonal decomposition (5.3) of $c^{\text{GR}},$ we easily obtain the decomposition $e={\Psi }_1-{\text{X}}_{12}{\widehat{\beta }}_x^{\prime }-{\text{X}}_1{\widehat{\beta }}_{x_1}^{\prime },$ where ${\widehat{\beta }}_x\mathrm{<mo>=</mo>}{\Psi }_1^{\prime }\Lambda {\text{X}}_{12}{\mathrm{<mo\; stretchy="false">(</mo>}{\text{X\hspace{0.17em}}}_{12}^{\prime }\Lambda {\text{X}}_{12}\mathrm{<mo\; stretchy="false">)</mo>}}^{-1}$ and ${\widehat{\beta }}_{x_1}={\Psi }_1^{\prime }\Lambda {\text{X}}_1{\mathrm{<mo\; stretchy="false">(</mo>}{\text{X\hspace{0.17em}}}_1^{\prime }\Lambda {\text{X}}_1\mathrm{<mo\; stretchy="false">)</mo>}}^{-1}$ are the coefficients of ${\widehat{t}}_x-{\tilde{t}}_x$ and $t_{x_1}-{\widehat{t}}_{x_1},$ respectively. Note that ${\Psi }_1-{\text{X}}_{12}{\widehat{\beta }}_x^{\prime }$ is the matrix of residuals in the GREG estimator ${\widehat{t}}_y^{\text{GR}|x}={\tilde{t}}_y+{\widehat{\beta }}_x\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{t}}_x-{\tilde{t}}_x\mathrm{<mo\; stretchy="false">)</mo>}$ resulting from calibration involving only ${\text{X}}_{12},$ and ${\Psi }_1-{\text{X}}_1{\widehat{\beta }}_{x_1}^{\prime }$ is the matrix of residuals in the GREG estimator ${\widehat{t}}_y^{\text{GR}|x_1}={\tilde{t}}_y+{\widehat{\beta }}_{x_1}\mathrm{<mo\; stretchy="false">(</mo>}{t}_{x_1}-{\widehat{t}}_{x_1}\mathrm{<mo\; stretchy="false">)</mo>}$ resulting from calibration involving only ${\text{X}}_1.$ Then, using the orthogonality of ${\text{X}}_1$ and ${\text{X}}_{12},$ it is shown without difficulty that

$$\widehat{\text{AV}}\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{t}}_y^{\text{GR}}\mathrm{<mo\; stretchy="false">)</mo>}-\widehat{\text{Var}}\mathrm{<mo\; stretchy="false">(</mo>}{\tilde{t}}_y\mathrm{<mo\; stretchy="false">)</mo>}=\widehat{\text{AV}}\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{t}}_y^{\text{GR}|x}\mathrm{<mo\; stretchy="false">)</mo>}-\widehat{\text{Var}}\mathrm{<mo\; stretchy="false">(</mo>}{\tilde{t}}_y\mathrm{<mo\; stretchy="false">)</mo>}+\widehat{\text{AV}}\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{t}}_y^{\text{GR}{|x}_1}\mathrm{<mo\; stretchy="false">)</mo>}-\widehat{\text{Var}}\mathrm{<mo\; stretchy="false">(</mo>}{\tilde{t}}_y\mathrm{<mo\; stretchy="false">)</mo>\; ,}$$

which implies that the reduction of variance due to using the two auxiliary variables $x_1$ and $x$ in the regression (also calibration) procedure is additive. Thus, recalling Remark 3.2, the generalized regression estimator retains this additivity property of the BLUE of $t_y.$

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