2. Basic theory

Jae-kwang Kim, Seunghwan Park and Seo-young Kim

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In this section, we first introduce the basic theory for combining the information for small area estimation. We first consider the simple case of combining two surveys. Assume that there are two surveys, survey $A$ and survey $B,$ obtained from separate probability sampling designs. The two surveys are not necessarily independent. From survey $A,$ we obtain a design unbiased estimator ${\widehat{X}}_{h\mathrm{,}a}={\displaystyle {\sum }_{i\in A_h}}{w}_{ia}{x}_i$ and its variance estimator $\widehat{V}\left({\widehat{X}}_h\right).$ From survey $B,$ we obtain a design unbiased estimator ${\widehat{Y}}_{1h}={\displaystyle {\sum }_{i\in B_h}}{w}_{ib}{y}_{1i}$ of $Y_{1h}={\displaystyle {\sum }_{i\in U_h}}{y}_{1i}.$ The sampling error of $\left({\widehat{X}}_h\mathrm{,}{\widehat{Y}}_{1h}\right)$ can be expressed by the sampling error model

$$\left(\begin{array}{l}{\widehat{X}}_h\hfill \\ {\widehat{Y}}_{1h}\hfill \end{array}\right)=\left(\begin{array}{l}{X}_h\hfill \\ Y_{1h}\hfill \end{array}\right)+\left(\begin{array}{l}{N}_h{a}_h\hfill \\ N_h{b}_h\hfill \end{array}\right)(2.1)$$

and $a_h$ and $b_h$ represent the sampling errors associated with ${\widehat{X}}_h/N_h$ and ${\widehat{Y}}_{1h}/N_h$ such that

$$\left(\begin{array}{c}{a}_h\\ b_h\end{array}\right)\sim \left[\left(\begin{array}{c}0\\ 0\end{array}\right)\mathrm{,}\left(\begin{array}{cc}V\left(a_h\right)& \text{Cov}\left(a_h\mathrm{,}{b}_h\right)\\ \text{Cov}\left(a_h\mathrm{,}{b}_h\right)& V\left(b_h\right)\end{array}\right)\right]\mathrm{.}$$

Our parameter of interest is the population total $X_h$ of $x$ in area $h.$

From (1.1), we obtain the following area level model:

$$Y_{1h}=N_h{\beta }_0+{\beta }_1{X}_h+{\tilde{e}}_{1h}\mathrm{,}(2.2)$$

where $\left(N_h\mathrm{,}{X}_h\mathrm{,}{Y}_{1h}\mathrm{,}{\tilde{e}}_{1h}\right)={\displaystyle {\sum }_{i\in U_h}}\left(\mathrm{1,}{x}_i\mathrm{,}{y}_{1i}\mathrm{,}{e}_{1i}\right).$ We can express (2.2) in terms of population mean

$${\overline{Y}}_{1h}={\beta }_0+{\overline{X}}_h{\beta }_1+{\overline{e}}_{1h}\mathrm{,}(2.3)$$

where $\left({\overline{X}}_h\mathrm{,}{\overline{Y}}_{1h}\mathrm{,}{\overline{e}}_{1h}\right)=N_h^{-1}{\displaystyle {\sum }_{i\in U_h}}\left(x_i\mathrm{,}{y}_{1i}\mathrm{,}{e}_{1i}\right).$ If we use a nested error model

$$e_{1hi}={\epsilon }_h+u_{hi}(2.4)$$

where ${\epsilon }_h\sim \left(\mathrm{0,}{\sigma }_e^2\right)$ and $u_{hi}\sim \left(\mathrm{0,}{\sigma }_u^2\right),$ then ${\overline{e}}_{1h}\sim \left(\mathrm{0,}{\sigma }_{e\mathrm{,}h}^2\right),$ ${\sigma }_{e\mathrm{,}h}^2={\sigma }_e^2+{\sigma }_u^2/N_h.$ The nested error model is quite popular in small area estimation (e.g., Battese, Harter and Fuller 1988) and it assumes that $\text{Cov}\left(e_{1hi}\mathrm{,}{e}_{1hj}\right)={\sigma }_e^2$ for $i\ne j.$ Because $N_h$ is often quite large, we can safely assume that ${\overline{e}}_{1h}\sim \left(\mathrm{0,}{\sigma }_{e\mathrm{,}h}^2={\sigma }_e^2\right).$ The model (2.2) is called structural error model because it describes the structural relationship between the two latent variables $Y_{1h}$ and $X_h.$ The two models, (2.1) and (2.2), are often encountered in the measurement error model literature (Fuller 1987). Thus, the model for small area estimation can be viewed as a measurement error model, as suggested by Fuller (1991) who originally used the measurement error model approach in the unit-level modeling for small area estimation.

Now, if we define $\left({\overline{y}}_{1h}\mathrm{,}{\overline{x}}_h\right)=N_h^{-1}\left({\widehat{Y}}_{1h}\mathrm{,}{\widehat{X}}_h\right),$ combining (2.1) and (2.3), we have

$$\left(\begin{array}{l}{\overline{y}}_{1h}\hfill \\ {\overline{x}}_h\hfill \end{array}\right)=\left(\begin{array}{ll}{\beta }_0\hfill & {\beta }_1\hfill \\ 0\hfill & 1\hfill \end{array}\right)\left(\begin{array}{l}1\hfill \\ {\overline{X}}_h\hfill \end{array}\right)+\left(\begin{array}{c}{b}_h+{\overline{e}}_{1h}\\ a_h\end{array}\right)$$

which can also be written as

$$\left(\begin{array}{l}{\overline{y}}_{1h}-{\beta }_0\hfill \\ {\overline{x}}_h\hfill \end{array}\right)=\left(\begin{array}{l}{\beta }_1\hfill \\ 1\hfill \end{array}\right){\overline{X}}_h+\left(\begin{array}{c}{b}_h+{\overline{e}}_{1h}\\ a_h\end{array}\right)\mathrm{.}(2.5)$$

Thus, when all the model parameters in (2.5) are known, the best estimator of ${\overline{X}}_h$ can be computed by

$${\widehat{\overline{X}}}_h={\left\{\left({\beta }_1\mathrm{,1}\right)V_h^{-1}{\left({\beta }_1\mathrm{,1}\right)}^{\prime }\right\}}^{-1}\left({\beta }_1\mathrm{,1}\right)V_h^{-1}{\left({\overline{y}}_{1h}-{\beta }_0\mathrm{,}{\overline{x}}_h\right)}^{\prime }(2.6)$$

where $V_h$ is the variance-covariance matrix of ${\left(b_h+{\overline{e}}_{1h}\mathrm{,}{a}_h\right)}^{\prime }.$ The variance of ${\widehat{\overline{X}}}_h$ is given by ${\left\{\left({\beta }_1\mathrm{,1}\right)V_h^{-1}{\left({\beta }_1\mathrm{,1}\right)}^{\prime }\right\}}^{-1}.$ The estimator in (2.6) can be called the Generalized Least Squares (GLS) estimator because it uses the technique of the generalized least squares method in the linear model theory. The GLS method is useful because it is optimal and it can incorporate additional sources of information naturally. For example, if another estimator ${\overline{y}}_{2h}$ for ${\overline{Y}}_{2h}$ is also available and satisfies

$${\overline{Y}}_{2h}={\gamma }_0+{\gamma }_1{\overline{X}}_h+{\overline{e}}_{2h}$$

and

$${\overline{y}}_{2h}={\overline{Y}}_{2h}+c_h\mathrm{,}$$

then the extended GLS model is written as

$$\left(\begin{array}{l}{\overline{y}}_{2h}-{\gamma }_0\hfill \\ {\overline{y}}_{1h}-{\beta }_0\hfill \\ {\overline{x}}_h\hfill \end{array}\right)=\left(\begin{array}{l}{\gamma }_1\hfill \\ {\beta }_1\hfill \\ 1\hfill \end{array}\right){\overline{X}}_h+\left(\begin{array}{c}{c}_h+{\overline{e}}_{2h}\\ b_h+{\overline{e}}_{1h}\\ a_h\end{array}\right)(2.7)$$

and the GLS estimator can be obtained by

$${\widehat{\overline{X}}}_{h2}={\left\{\left({\gamma }_1\mathrm{,}{\beta }_1\mathrm{,1}\right)V_{h2}^{-1}{\left({\gamma }_1\mathrm{,}{\beta }_1\mathrm{,1}\right)}^{\prime }\right\}}^{-1}\left({\gamma }_1\mathrm{,}{\beta }_1\mathrm{,1}\right)V_{h2}^{-1}{\left({\overline{y}}_{2h}-{\gamma }_0\mathrm{,}{\overline{y}}_{1h}-{\beta }_0\mathrm{,}{\overline{x}}_h\right)}^{\prime }$$

where $V_{h2}$ is the variance-covariance matrix of ${\left(c_h+{\overline{e}}_{2h}\mathrm{,}{b}_h+{\overline{e}}_{1h}\mathrm{,}{a}_h\right)}^{\prime }.$ The GLS estimator has variance ${\left\{\left({\gamma }_1\mathrm{,}{\beta }_1\mathrm{,1}\right)V_{h2}^{-1}{\left({\gamma }_1\mathrm{,}{\beta }_1\mathrm{,1}\right)}^{\prime }\right\}}^{-1}.$ If ${\overline{y}}_{2h}$ is independent of $\left({\overline{x}}_h\mathrm{,}{\overline{y}}_{1h}\right),$ the efficiency gain by incorporating ${\overline{y}}_{2h}$ into GLS in terms of relative variance can be expressed as

$$\frac{V\left({\widehat{\overline{X}}}_{h2}\right)-V\left({\widehat{\overline{X}}}_h\right)}{V\left({\widehat{\overline{X}}}_h\right)}=-\frac{{\left\{V\left({\overline{y}}_{2h}/{\gamma }_1\right)\right\}}^{-1}}{{\left\{V\left({\widehat{\overline{X}}}_h\right)\right\}}^{-1}+{\left\{V\left({\overline{y}}_{2h}/{\gamma }_1\right)\right\}}^{-1}}\mathrm{,}$$

where $V\left({\overline{y}}_{2h}/{\gamma }_1\right)=V\left(c_h+{\overline{e}}_{2h}\right)/{\gamma }_1^2.$ The gain is high if both the sampling variance of ${\overline{y}}_{2h}$ and the model variance $V\left({\overline{e}}_{2h}\right)$ are small. If ${\gamma }_1=0,$ then there is no gain.

Remark 1 Note that model (2.5) can also be written as

$$\left(\begin{array}{c}{\beta }_1^{-1}\left({\overline{y}}_{1h}-{\beta }_0\right)\\ {\overline{x}}_h\end{array}\right)=\left(\begin{array}{l}1\hfill \\ 1\hfill \end{array}\right){\overline{X}}_h+\left(\begin{array}{c}\left(b_h+{\overline{e}}_{1h}\right)/{\beta }_1\\ a_h\end{array}\right)\mathrm{.}(2.8)$$

The GLS estimator obtained from (2.8), which is the same as the GLS estimator obtained from (2.5), can be expressed as

$${\widehat{\overline{X}}}_h={\alpha }_h{\overline{x}}_h+\left(1-{\alpha }_h\right){\tilde{x}}_h(2.9)$$

where ${\tilde{x}}_h={\beta }_1^{-1}\left({\overline{y}}_{1h}-{\beta }_0\right)$ and

$$\begin{array}{lll}{\alpha }_h\hfill & =\hfill & \frac{V\left({\tilde{x}}_h\right)-\text{Cov}\left({\overline{x}}_h\mathrm{,}{\tilde{x}}_h\right)}{V\left({\overline{x}}_h\right)+V\left({\tilde{x}}_h\right)-2\text{Cov}\left({\overline{x}}_h\mathrm{,}{\tilde{x}}_h\right)}\hfill \\ \hfill & =\hfill & \frac{{\sigma }_{e\mathrm{,}h}^2+V\left(b_h\right)-{\beta }_1\text{Cov}\left(a_h\mathrm{,}{b}_h\right)}{{\sigma }_{e\mathrm{,}h}^2+V\left(b_h\right)+{\beta }_1^{2}V\left(a_h\right)-2{\beta }_1\text{Cov}\left(a_h\mathrm{,}{b}_h\right)}.\hfill \end{array}$$

The estimator ${\tilde{x}}_h,$ when computed with estimated parameter   $\widehat{\beta }=\left({\widehat{\beta }}_0\mathrm{,}{\widehat{\beta }}_1\right),$ is called the synthetic estimator and the optimal estimator in (2.9) is often called the composite estimator. It can be shown that, ignoring the effect of estimating $\beta ,$ the variance of the composite estimator is equal to

$$V\left({\widehat{\overline{X}}}_h-{\overline{X}}_h\right)={\alpha }_{h}V\left({\overline{x}}_h\right)+\left(1-{\alpha }_h\right)\text{Cov}\left({\overline{x}}_h\mathrm{,}{\tilde{x}}_h\right)(2.10)$$

and, as ${\alpha }_h<1,$ the composite estimator is more efficient than the direct estimator.

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