3. Composite generalized regression estimation for design (c)

Takis Merkouris

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A computationally very convenient, but generally suboptimal, variant of ${\widehat{ℬ}}^o$ in (2.6) is obtained by replacing the matrix ${\Lambda }^0$ with the diagonal "weighting matrix� $\Lambda$ having $w_{ik}/q_{ik}$ as $i{k}^{\text{th}}$ diagonal entry, where $\left\{w_{ik}\right\}$ are the design weights of $S_i$ and $\left\{q_{ik}\right\}$ are positive constants. This gives the multivariate composite generalized regression (CGR) estimator of ${\left({t^{\prime }}_x\mathrm{,}{t^{\prime }}_y\right)}^{\prime }$

$$\left(\begin{array}{c}{\widehat{X}}^{\text{CGR}}\\ {\widehat{Y}}^{\text{CGR}}\end{array}\right)=\widehat{ℬ}\left(\begin{array}{c}{\widehat{X}}_1\\ {\widehat{Y}}_2\end{array}\right)+(I-\widehat{ℬ})\left(\begin{array}{c}{\widehat{X}}_3\\ {\widehat{Y}}_3\end{array}\right)=\left(\begin{array}{c}{\widehat{X}}_3\\ {\widehat{Y}}_3\end{array}\right)+\widehat{ℬ}\left(\begin{array}{c}{\widehat{X}}_1-{\widehat{X}}_3\\ {\widehat{Y}}_2-{\widehat{Y}}_3\end{array}\right)\mathrm{,}(3.1)$$

where $\widehat{ℬ}=({{\mathcal{X}}^{\prime }}_3\Lambda \mathcal{X}){({\mathcal{X}}^{\prime }\Lambda \mathcal{X})}^{-1}$ is the associated matrix regression coefficient. For an extensive discussion of the generalized regression estimator in a single sample, see Särndal et al. (1992, Chapter 6). The CGR estimator may be compactly written as ${\widehat{\mathcal{X}}}^{\text{CGR}}={\widehat{\mathcal{X}}}_3-\widehat{ℬ}\widehat{\mathcal{X}}\left[={\left({\mathcal{X}}_3-\mathcal{X}{\widehat{ℬ}}^{\prime }\right)}^{\prime }w\right],$ i.e., as a sum of weighted sample regression residuals. The coefficient $\widehat{ℬ}$ is optimal in the sense of generalized least squares, i.e., it minimizes the quadratic form ${\left({\mathcal{X}}_3-\mathcal{X}{\widehat{ℬ}}^{\prime }\right)}^{\prime }\Lambda \left({\mathcal{X}}_3-\mathcal{X}{\widehat{ℬ}}^{\prime }\right)$ in these residuals. Similarly to the COR estimator, the CGR estimator too can be obtained in calibration form as ${{\mathcal{X}}^{\prime }}_{3}c,$ where the vector $c=w+\Lambda \mathcal{X}{\left({\mathcal{X}}^{\prime }\Lambda \mathcal{X}\right)}^{-1}\left(0-{\mathcal{X}}^{\prime }w\right)$ minimizes the generalized least-squares distance ${\left(c-w\right)}^{\prime }{\Lambda }^{-1}$ $\left(c-w\right)$ and satisfies the constraints ${\widehat{X}}_1^{\text{CGR}}={\widehat{X}}_3^{\text{CGR}}$ and ${\widehat{Y}}_2^{\text{CGR}}={\widehat{Y}}_3^{\text{CGR}}.$ This extends to the present context the well-known equivalence of generalized regression estimation and calibration estimation (Deville and Särndal 1992) for a single-sample setting. Now using the subvector of calibrated weights $c_3,$ for sample $S_3$ only, we obtain the composite estimators in (3.1) in the simple linear forms ${\widehat{X}}^{\text{CGR}}={X^{\prime }}_3{c}_3$ and ${\widehat{Y}}^{\text{CGR}}={Y^{\prime }}_3{c}_3.$ Using Lemma 1 and the diagonal structure of $\Lambda ,$ it works out that ${\widehat{X}}^{\text{CGR}}$ can be written as

$${\widehat{X}}^{\text{CGR}}={\widehat{B}}_{1x}{\widehat{X}}_1+\left(I-{\widehat{B}}_{1x}\right){\widehat{X}}_3^{\text{GR}}\mathrm{,}(3.2)$$

where ${\widehat{X}}_3^{\text{GR}}={\widehat{X}}_3+{X^{\prime }}_3\Lambda \Psi {\left({\Psi }^{\prime }\Lambda \Psi \right)}^{-1}\left({\widehat{Y}}_2-{\widehat{Y}}_3\right)$ is the generalized regression (GR) counterpart of ${\widehat{X}}_3^{\text{OR}}.$ The matrix regression coefficient ${\widehat{B}}_{1x}$ is written explicitly as ${\widehat{B}}_{1x}={X^{\prime }}_3{L}_{\Psi }X{\left({X^{\prime }}_1{\Lambda }_1{X}_1+{X^{\prime }}_3{L}_{\Psi }X\right)}^{-1},$ where ${X^{\prime }}_3{L}_{\Psi }X={X^{\prime }}_3{\Lambda }_3{X}_3-{X^{\prime }}_3{\Lambda }_3{Y}_3{\left({Y^{\prime }}_2{\Lambda }_2{Y}_2+{Y^{\prime }}_3{\Lambda }_3{Y}_3\right)}^{-1}{Y^{\prime }}_3{\Lambda }_3{X}_3.$ If $x$ and $y$ were uncorrelated, or if information on $y$ was not used in the estimation of $t_x,$ then it would be ${\widehat{X}}_3^{\text{GR}}={\widehat{X}}_3$ and ${\widehat{B}}_{1x}={X^{\prime }}_3{\Lambda }_3{X}_3{\left({X^{\prime }}_1{\Lambda }_1{X}_1+{X^{\prime }}_3{\Lambda }_3{X}_3\right)}^{-1}.$ But the GR estimator ${\widehat{X}}_3^{\text{GR}}$ is generally more efficient than the HT estimator ${\widehat{X}}_3,$ and since ${X^{\prime }}_1{\Lambda }_1{X}_1+{X^{\prime }}_3{L}_{\Psi }X\text{<}{X^{\prime }}_1{\Lambda }_1{X}_1+{X^{\prime }}_3{\Lambda }_3{X}_3$ (in the partial ordering of non-negative definite matrices), it is clear that more weight is given to ${\widehat{X}}_3^{\text{GR}}$ in (3.2), through $I-{\widehat{B}}_{1x}={X^{\prime }}_1{\Lambda }_1{X}_1{\left({X^{\prime }}_1{\Lambda }_1{X}_1+{X^{\prime }}_3{L}_{\Psi }X\right)}^{-1},$ than would have been given to the component estimator ${\widehat{X}}_3$ in the simple composite estimator involving only information on $x.$ This suggests that the CGR estimator in (3.2), incorporating information from sample $S_2,$ is a more efficient estimator. Suggestive of the efficiency of ${\widehat{X}}^{\text{CGR}}$ is also its alternative expression, obtained using (2.11), ${\widehat{X}}^{\text{CGR}}={\tilde{X}}^{\text{CGR}}+{X^{\prime }}_3{L}_X\Psi {\left({\Psi }^{\prime }{L}_X\Psi \right)}^{-1}\left[{\widehat{Y}}_2-{\widehat{Y}}_3^{\text{GR}}\right],$ where ${\tilde{X}}^{\text{CGR}}={\widehat{X}}_3+{X^{\prime }}_3\Lambda X{\left(X^{\prime }\Lambda X\right)}^{-1}\left({\widehat{X}}_1-{\widehat{X}}_3\right)={\tilde{B}}_{1x}{\widehat{X}}_1+\left(I-{\tilde{B}}_{1x}\right){\widehat{X}}_3$ is the composite regression estimator of $t_x$ using information on $x$ from $S_1$ and $S_3.$

In general, the computationally simpler CGR estimator $\left({\widehat{X}}^{\text{CGR}}\mathrm{,}{\widehat{Y}}^{\text{CGR}}\right),$ involving the coefficient $\widehat{ℬ},$ is less efficient than the optimal composite regression estimator $\left({\widehat{X}}^{\text{COR}}\mathrm{,}{\widehat{Y}}^{\text{COR}}\right)$ which involves the estimated optimal coefficient ${\widehat{ℬ}}^o$ and has the same asymptotic variance as the BLUE in (2.3); the efficiency loss may be larger in nested matrix sampling, for which the matrix ${\Lambda }^0$ is not block-diagonal. On the other hand, $\left({\widehat{X}}^{\text{COR}}\mathrm{,}{\widehat{Y}}^{\text{COR}}\right)$ may be unstable in small samples, when there is a small number of degrees of freedom available for the estimation of ${\widehat{ℬ}}^o,$ which is particularly so in nested matrix sampling; for a discussion of the relative stability of the optimal versus the generalized regression estimator in the single-sample case see Rao (1994) or Montanari (1998). For certain sampling strategies, described in the following theorem, $\widehat{ℬ}={\widehat{ℬ}}^o$ and the CGR estimator is the COR estimator, and asymptotically is BLUE; the proof is given in the Appendix.

Theorem 1 Consider the following sampling strategies.

Non-nested design

• $\left(a\right)$ For all three samples $S_1,S_2$ and $S_3$ assume stratified simple random sampling without replacement (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 $q_{ik}$ in ${\Lambda }_i$ as $q_{ik}=\left(n_{ih}-1\right)/N_{ih}\left(1-f_{ih}\right)$ for all units of stratum $h.$ Furthermore, assume that within each sample the units are sorted by stratum, and consider the augmented design matrix $\mathcal{Z}=\left(\mathcal{X}\mathrm{,}D\right)$ in (2.7), where $D$ is the block diagonal matrix $\text{diag}\left\{D_1\mathrm{,}{D}_2\mathrm{,}{D}_3\right\}$ and $D_i$ is the diagonal matrix $\text{diag}\left\{1_{i1}\mathrm{,}\dots \mathrm{,}{1}_{ih}\mathrm{,}\dots \mathrm{,}{1}_{i{H}_i}\right\},$ with diagonal element $1_{ih}$ being a vector of ones for all units of stratum $h$ in sample $S_i,$ and consider the corresponding augmented vector of calibration totals $t_{\mathcal{Z}}={\left(0^{\prime }\mathrm{,}{0}^{\prime }\mathrm{,}{N^{\prime }}_1\mathrm{,}{N^{\prime }}_2\mathrm{,}{N^{\prime }}_3\right)}^{\prime },$ where $N_i$ is the vector of strata sizes for sample $S_i.$
• $\left(b\right)$  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 $h,$ where ${\pi }_{ihk}$ is the inclusion probability of unit $k$ in stratum $h$ of the $i^{\text{th}}$ survey.

Nested design

• $\left(a’\right)$ Assume that an initial stratified simple random sample $S$ is split by stratum into three simple random subsamples $S_1,S_2$ and $S_3.$ Specify the sampling fractions $f_{ih},$ the constants $q_{ik}$ in ${\Lambda }_i,$ the design matrix $\mathcal{Z}=\left(\mathcal{X}\mathrm{,}D\right)$ and the vector of calibration totals $t_{\mathcal{Z}}$ as in part $\left(a\right).$
• $\left(b’\right)$ Assume that an initial stratified Poisson sample $S$ is randomly split by stratum into three subsamples $S_1,S_2$ and $S_3,$ with unequal inclusion probabilities for the units of each subsample. Specify the constants $q_{ik}$ in ${\Lambda }_i$ as $q_{ik}={\pi }_{ihk}/\left(1-{\pi }_{ihk}\right)$ for the units of stratum $h,$ where ${\pi }_{ihk}$ is the marginal inclusion probability of unit $k$ in stratum $h$ of the $i^{\text{th}}$ subsample.

Under each of strategies $\left(a\right)$ and $\left(b\right),$ the calibration procedure with matrix $\Lambda$ in the least-squares distance measure gives the CGR estimator in (3.1) with $\widehat{ℬ}={\widehat{ℬ}}^o,$ implying that the CGR estimator is the COR estimator. For $\left(a’\right)$ and $\left(b’\right),$ this holds approximately when the strata sampling fractions are approximately zero.

Corollary 1 The result of Theorem 1 holds also for the unstratified versions of all four designs. For simple random sampling without replacement (SRS), in particular, the matrix $D$ reduces to the diagonal matrix $\text{diag}\left\{1_1\mathrm{,}{1}_2\mathrm{,}{1}_3\right\}$ having as its $i^{\text{th}}$ diagonal element the $n_i-$ dimensional unit vector $1_i,$ and the vector of calibration totals is then $t_{\mathcal{Z}}={\left(0^{\prime }\mathrm{,}{0}^{\prime }\mathrm{,}N\mathrm{,}N\mathrm{,}N\right)}^{\prime }.$

Corollary 2 In non-nested sampling, when the sampling design for each of the three samples is one of the designs in $\left(a\right)$ and $\left(b\right)$ or one of their unstratified versions, but not the same for all samples, the result of Theorem 1 holds provided that the matrix $D$ in $\mathcal{Z}$and the vector $t_{\mathcal{Z}}$ are reduced so as to correspond only to the samples for which SRS or STRSRS is used.

The extended calibration scheme in Theorem $1\left(a,a’\right)$ includes calibration to the stratum sizes (or to the population size in the SRS version), through the inclusion of an intercept for each stratum in the design matrix $\mathcal{X}.$ No additional information is used beyond what is assumed in the sampling design in $\left(a\right)$ and $\left(a’\right),$ and the form of the resulting CGR estimator remains the same as in (3.1) because the HT estimates of the population and strata sizes are exact. The effect of this extended calibration (with the specified values of $q_{ik})$ is only to convert the CGR coefficient $\widehat{ℬ}$ to the optimal coefficient ${\widehat{ℬ}}^o$ and, thus, the CGR estimator to the COR estimator. The practical significance of this conversion lies in carrying out optimal composite regression estimation through the much simpler calibration procedure of generalized regression estimation.

Subsampling as in part $\left(a’\right),$ with a priori fixed sample sizes, is a natural procedure in matrix sampling involving splitting a questionnaire. In contrast, in the subsampling scheme of part $\left(b’\right)$ $n_i$ is the expected sample size of $S_i,$ the actual size being random. Unequal subsampling probabilities may be determined adaptively for increased efficiency; see Gonzalez and Eltinge (2008).

The results of Theorem 1 could extend to other sampling designs, e.g., stratified two-stage simple random sampling in non-nested matrix sampling. However, the required adjustments in the matrices ${\Lambda }_i$ would not be easier than using directly the matrices ${\Lambda }_i^0$ in the calibration to obtain the optimal composite regression estimator.

For sampling designs other than those assumed in Theorem 1, the value of $q_{ik}$ in the entries of ${\Lambda }_i$ should be set to $q_{ik}={\tilde{n}}_i/\left({\tilde{n}}_1+{\tilde{n}}_2+{\tilde{n}}_3\right),$ where ${\tilde{n}}_i=n_i/d_i,d_i$ denoting design effect, to take into account the differential in effective sample sizes among the three samples. If the same design is used for all samples, then ${\tilde{n}}_i=n_i.$ The justification for this adjustment is based on the argument given in Merkouris (2010) for a similar problem of composite regression estimation.

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