5. Domain estimation

Takis Merkouris

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Composite estimators for domains (subpopulations) of interest may be readily obtained using the calibrated weights derived in the previous sections, that is, by summing the weighted values of a variable over any domain $U_d\subset U.$ For instance, letting $X_{id}$ denote the matrix $X_i,$ for sample $S_i,$ with the entries of the $k^{\text{th}}$ row set equal to 0 if $k\notin U_d,$ the CGR estimator of the domain total $t_{xd}$ based on the weights of $S_3$ calibrated with the scheme of design (c) (see Section 3) is given by

$${\widehat{X}}_{3d}^{\text{CGR}}={X^{\prime }}_{3d}{c}_3={\widehat{X}}_{3d}^{\text{GR}}+{X^{\prime }}_{3d}{L}_{\Psi }X{\left(X^{\prime }{L}_{\Psi }X\right)}^{-1}\left[{\widehat{X}}_1-{\widehat{X}}_3^{\text{GR}}\right]\mathrm{,}$$

where ${\widehat{X}}_{3d}^{\text{GR}}={\widehat{X}}_{3d}+{X^{\prime }}_{3d}\Lambda \Psi {\left({\Psi }^{\prime }\Lambda \Psi \right)}^{-1}\left({\widehat{Y}}_2-{\widehat{Y}}_3\right)$ and the subscript $d$ indicates domain. The CGR estimator ${\widehat{X}}_{1d}^{\text{CGR}}$ based on sample $S_1$ is obtained in the same manner. However, unlike the population-level estimator (3.2), resulting from calibration of two estimators to each other at population level, the estimators ${\widehat{X}}_{1d}^{\text{CGR}}$ and ${\widehat{X}}_{3d}^{\text{CGR}}$ are not constructed as composites of two domain estimators, based on samples $S_1$ and $S_3,$ and they are not identical. Moreover, although both ${\widehat{X}}_{1d}^{\text{CGR}}$ and ${\widehat{X}}_{3d}^{\text{CGR}}$ incorporate information on $x$ from samples $S_1$ and $S_3,$ their construction (non-customized at domain level) may entail some loss of efficiency.

A simple modification of the calibration procedure that leads to efficient composite estimation for all totals of interest involves the augmentation of the design matrix with columns defined at each domain level for the relevant variables. Thus, for design (c) estimation of the domain total $t_{xd}$ involves the augmentation of the design matrix $\mathcal{X}$ in (2.7) with the column ${\left(-{X^{\prime }}_{1d}\mathrm{,}{0}^{\prime }\mathrm{,}{X^{\prime }}_{3d}\right)}^{\prime }.$ The resulting estimator, ${\tilde{X}}_d^{\text{CGR}},$ may be written in the forms

$$\begin{array}{lll}{\tilde{X}}_d^{\text{CGR}}\hfill & =\hfill & {\widehat{X}}_{3d}+{\widehat{B}}_{1xd}\left({\widehat{X}}_1-{\widehat{X}}_3\right)+{\widehat{B}}_{2xd}\left({\widehat{Y}}_2-{\widehat{Y}}_3\right)+{\widehat{B}}_{3xd}\left({\widehat{X}}_{1d}-{\widehat{X}}_{3d}\right)\hfill \\ \hfill & =\hfill & {\widehat{B}}_{1xd}{\tilde{X}}_{1d}^{\text{GR}}+\left(I-{\widehat{B}}_{1xd}\right){\tilde{X}}_{3d}^{\text{GR}}\mathrm{,}\hfill \end{array}(5.1)$$

where ${\tilde{X}}_{1d}^{\text{GR}}$ and ${\tilde{X}}_{3d}^{\text{GR}}$ are now the GR domain estimators incorporating the regression effect of the second and third terms of (5.1). Adding another term in (5.1) involving the difference ${\widehat{Y}}_{2d}-{\widehat{Y}}_{3d}$ may not improve appreciably the efficiency of ${\tilde{X}}_d^{\text{CGR}}$ but will be necessary if estimation of the domain total $t_{yd}$ is also required. In any particular situation, the augmentation of the design matrix $\mathcal{X}$ involves only those components of $x$ or $y$ for which domain estimates are needed. A possible drawback of this procedure is the additional computational burden, which increases with the number of domains and the variables for which domain estimation is required.

An alternative approach that may be more appropriate when the domain estimates of interest are numerous, involves the separate production of the domain estimates by carrying out the composite calibration only at the domain level. For the domain total $t_{xd},$ this would give the domain CGR estimator, in analogy with the population CGR estimator (3.2),

$${\stackrel{⌣}{X}}_d^{\text{CGR}}={\stackrel{⌣}{B}}_{1xd}{\widehat{X}}_{1d}+\left(I-{\stackrel{⌣}{B}}_{1xd}\right){\stackrel{⌣}{X}}_{3d}^{\text{GR}}\mathrm{,}$$

where ${\stackrel{⌣}{B}}_{1xd}={X^{\prime }}_{3d}{L}_{{\Psi }_d}{X}_d{\left({X^{\prime }}_d{L}_{{\Psi }_d}X\right)}_d^{-1}$ and ${\stackrel{⌣}{X}}_{3d}^{\text{GR}}={\widehat{X}}_{3d}+{X^{\prime }}_{3d}\Lambda {\Psi }_d{\left({{\Psi }^{\prime }}_d\Lambda {\Psi }_d\right)}^{-1}\left({\widehat{Y}}_{2d}-{\widehat{Y}}_{3d}\right).$ The efficiency of the joint estimator $\left({\stackrel{⌣}{X}}_d^{\text{CGR}}\mathrm{,}{\stackrel{⌣}{Y}}_d^{\text{CGR}}\right)$ over the estimator $\left({\widehat{X}}_d^{\text{CGR}}\mathrm{,}{\widehat{Y}}_d^{\text{CGR}}\right)$ can be verified under the conditions of the following proposition (its proof in the Appendix).

Proposition 2 Under the sampling schemes of Theorem 1,

$$\widehat{\text{AV}}\left(\begin{array}{c}{\stackrel{⌣}{X}}_{3d}^{\text{CGR}}\\ {\stackrel{⌣}{Y}}_{3d}^{\text{GGR}}\end{array}\right)\text{<}\widehat{\text{AV}}\left(\begin{array}{c}{\widehat{X}}_{3d}^{\text{CGR}}\\ {\widehat{Y}}_{3d}^{\text{CGR}}\end{array}\right)\mathrm{.}$$

Notably, the drawback of a separate production of the domain estimates, through composite calibration at the domain level, is the loss of consistency among estimates at population level and domain level.

The above considerations extend to domain estimation for matrix sampling design (d).

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