Model-based small area estimation under informative sampling 3. Proposed methodModel-based small area estimation under informative sampling 3. Proposed method

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The proposed method of estimating the small area means, ${\overline{Y}}_i,$ is simple. It uses the standard EBLUP estimator under the augmented sample model (1.4). The model parameters $\left({\sigma }_{v0}^2,{\sigma }_{e0}^2\right)$ and $\left({\beta }_0,{\delta }_0\right)$ are estimated by REML and weighted least squares (WLS) respectively. The EBLUP estimator of ${\mu }_i$ under the augmented model (1.4) is given by

$${\widehat{\mu }}_{i\left(a\right)}^H={\widehat{\gamma }}_{i0}{\overline{y}}_i+{\left({\overline{X}}_i-{\widehat{\gamma }}_{i0}{\overline{x}}_i\right)}^T{\widehat{\beta }}_0+\left({\overline{G}}_i-{\widehat{\gamma }}_{i0}{\overline{g}}_i\right){\widehat{\delta }}_0,(3.1)$$

where ${\widehat{\gamma }}_{i0}={\widehat{\sigma }}_{v0}^2/\left({\widehat{\sigma }}_{v0}^2+{\widehat{\sigma }}_{e0}^2/n_i\right),\left({\widehat{\beta }}_0^T,{\widehat{\delta }}_0\right)$ is the WLS estimator of $\left({\beta }_0^T,{\delta }_0\right)$ and ${\overline{g}}_i={\displaystyle {\sum }_{j=1}^{n_i}{g}_{ij}}/n_i.$ Note that ${\widehat{\mu }}_{i\left(a\right)}^H$ assumes that ${\overline{G}}_i$ is known. The EBLUP estimator of ${\overline{Y}}_i$ under the augmented model may be written in terms of ${\widehat{\mu }}_{i\left(a\right)}^{\text{H}}$ as

$${\widehat{\overline{Y}}}_{i\left(a\right)}^H=N_i^{-1}\left[\left(N_i-n_i\right){\widehat{\mu }}_{i\left(a\right)}^{\text{H}}+n_i\left\{{\overline{y}}_i+{\left({\overline{X}}_i-\overline{x}{}_i\right)}^T{\widehat{\beta }}_0+\left({\overline{G}}_i-\overline{g}{}_i\right){\widehat{\delta }}_0\right\}\right].(3.2)$$

The pseudo-EBLUP estimator of ${\mu }_i$ under the augmented model (1.4) is similarly obtained by modifying (3.1) as

$${\widehat{\mu }}_{i\left(a\right)}^{\text{YR}}={\widehat{\gamma }}_{i0w}{\overline{y}}_{iw}+{\left({\overline{X}}_i-{\widehat{\gamma }}_{i0w}{\overline{x}}_{iw}\right)}^T{\widehat{\beta }}_{0w}+\left({\overline{G}}_i-{\widehat{\gamma }}_{i0w}{\overline{g}}_{iw}\right){\widehat{\delta }}_{0w},(3.3)$$

where ${\widehat{\gamma }}_{i0w}={\widehat{\sigma }}_{v0}^2/\left({\widehat{\sigma }}_{v0}^2+{\delta }_i^2{\widehat{\sigma }}_{e0}^2\right),{\overline{g}}_{iw}={\displaystyle {\sum }_{j=1}^{n_i}{\tilde{w}}_{j|i}{g}_{ij}}$ and $\left({\widehat{\beta }}_{0w},{\widehat{\delta }}_{0w}\right)$ are obtained by suitably modifying (2.5).

The MSE estimators of ${\widehat{\mu }}_{i\left(a\right)}^H$ and ${\widehat{\mu }}_{i\left(a\right)}^{\text{YR}}$ under the augmented model (1.4) are obtained by suitably modifying (2.8) and (2.9) respectively. Note that we only need to apply existing formulae to the augmented sample model (1.4) to get the EBLUP and the pseudo-EBLUP estimators and associated MSE estimators. New software development is not needed.

Our main interest is to study the performance of the estimators of ${\overline{Y}}_i$ based on the sample augmented model under informative sampling. Since the estimators ${\widehat{\overline{Y}}}_{i\left(a\right)}^H$ and ${\widehat{\mu }}_{i\left(a\right)}^{\text{YR}}$ are obtained under the augmented model (1.4), they are likely to perform well for the following reasons: (a) If the augmented model holds for the sample, then it also holds for the population, and the non-sampled values $y_{ij}$ can be predicted by fitting the augmented model to the sample; (b) If the augmenting variable $g_{ij}$ explains $y_{ij}$ after conditioning on $x_{ij},$ then ${\sigma }_{e0}^2$ and ${\sigma }_{v0}^2$ may be smaller than the corresponding ${\sigma }_e^2$ and ${\sigma }_v^2$ for the original population model, thus leading to better predictors of the non-sampled $y_{ij}.$ Pfeffermann and Sverchkov (2003) demonstrated, under a different model setup, that the inclusion of sample selection probabilities in the model “can reduce the RMSE quite substantially”.

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