Model-based small area estimation under informative sampling 2. Existing methodsModel-based small area estimation under informative sampling 2. Existing methods

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2.1 Estimators of small area means

Suppose that the population model (1.1) holds for the sample. Then the EBLUP estimator of ${\mu }_i={\overline{X}}_i^T\beta +v_i$ is given by

$${\widehat{\mu }}_i^H={\overline{X}}_i^T\widehat{\beta }+{\widehat{v}}_i={\widehat{\gamma }}_i{\overline{y}}_i+{\left({\overline{X}}_i-{\widehat{\gamma }}_i{\overline{x}}_i\right)}^T\widehat{\beta },(2.1)$$

where ${\widehat{\gamma }}_i={\widehat{\sigma }}_v^2/\left({\widehat{\sigma }}_v^2+{\widehat{\sigma }}_e^2/n_i\right),{\overline{y}}_i={\displaystyle {\sum }_{j=1}^{n_i}{y}_{ij}}/n_i,{\overline{x}}_i={\displaystyle {\sum }_{j=1}^{n_i}{x}_{ij}}/n_i$ are the unweighted sample means of the response variable $y$ and the covariates $x$ and ${\widehat{v}}_i={\widehat{\gamma }}_i\left({\overline{y}}_i-{\overline{x}}_i^T\widehat{\beta }\right).$ Further,

$$\widehat{\beta }={\left\{{\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^{n_i}{x}_{ij}{\left(x_{ij}-{\widehat{\gamma }}_i{\overline{x}}_i\right)}^T}}\right\}}^{-1}\left\{{\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^{n_i}\left(x_{ij}-{\widehat{\gamma }}_i{\overline{x}}_i\right)y_{ij}}}\right\}(2.2)$$

and ${\widehat{\sigma }}_e^2$ and ${\widehat{\sigma }}_v^2$ are obtained by the method of fitting of constants (Battese, Harter and Fuller (1988); Rao (2003, Chapter 7)) or restricted maximum likelihood (REML). The EBLUP estimator of the area mean ${\overline{Y}}_i$ may be written in terms of ${\widehat{\mu }}_i^H$ as

$${\widehat{\overline{Y}}}_i^H=N_i^{-1}\left[\left(N_i-n_i\right){\widehat{\mu }}_i^H+n_i\left\{{\overline{y}}_i+{\left({\overline{X}}_i-{\overline{x}}_i\right)}^T\widehat{\beta }\right\}\right],(2.3)$$

(see Rao 2003, page 141). Note that ${\widehat{\overline{Y}}}_i^H\approx {\widehat{\mu }}_i^H$ if the sampling fraction $n_i/N_i$ is sufficiently small. The EBLUP estimator ${\widehat{\overline{Y}}}_i^H$ is design consistent under simple random sampling (SRS) or stratified SRS with proportional allocation within area $i,$ leading to equal ${\pi }_{j|i}.$

The pseudo-EBLUP estimator of ${\mu }_i$ is given by

$${\widehat{\mu }}_i^{\text{YR}}={\widehat{\gamma }}_{iw}{\overline{y}}_{iw}+{\left({\overline{X}}_i-{\widehat{\gamma }}_{iw}{\overline{x}}_{iw}\right)}^T{\widehat{\beta }}_w,(2.4)$$

where we denote by ${\tilde{w}}_{j|i}=w_{j|i}/{\displaystyle {\sum }_{k=1}^{n_i}{w}_{k|i}}$ the normalized weights, ${\widehat{\gamma }}_{iw}={\widehat{\sigma }}_v^2/\left({\widehat{\sigma }}_v^2+{\delta }_i^2{\widehat{\sigma }}_e^2\right)$ with ${\delta }_i^2={\displaystyle {\sum }_{j=1}^{n_i}{\tilde{w}}_{j|i}^2},{\overline{y}}_{iw}={\displaystyle {\sum }_{j=1}^{n_i}{\tilde{w}}_{j|i}{y}_{ij}},{\overline{x}}_{iw}={\displaystyle {\sum }_{j=1}^{n_i}{\tilde{w}}_{j|i}{x}_{ij}}$ are the $i^{\text{th}}$ area weighted means of $y$ and $x,$ and

$${\widehat{\beta }}_w={\left[{\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^{n_i}{w}_{j|i}{x}_{ij}{\left(x_{ij}-{\widehat{\gamma }}_{iw}{\overline{x}}_{iw}\right)}^T}}\right]}^{-1}\left[{\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^{n_i}{w}_{j|i}\left(x_{ij}-{\widehat{\gamma }}_{iw}{\overline{x}}_{iw}\right)y_{ij}}}\right].(2.5)$$

The pseudo-EBLUP estimator ${\widehat{\mu }}_i^{\text{YR}}$ is design consistent under arbitrary selection probabilities $p_{j|i}$ unlike the EBLUP ${\widehat{\overline{Y}}}_i^H.$

Pfeffermann and Sverchkov (2007) studied estimation of small area means under informative sampling, assuming model (1.3) for the sample data and model (1.2) for the weights $w_{j|i}.$ Under this assumption, they obtained an estimator of ${\overline{Y}}_i$ that provides protection against informative sampling. It is given by

$${\widehat{\overline{Y}}}_i^{\text{PS}}=N_i^{-1}\left[\left(N_i-n_i\right){\widehat{\mu }}_{iu}^H+n_i\left\{{\overline{y}}_i+{\left({\overline{X}}_i-{\overline{x}}_i\right)}^T\widehat{\alpha }\right\}+\left(N_i-n_i\right)\widehat{b}{\widehat{\sigma }}_h^2\right],(2.6)$$

where ${\widehat{\mu }}_{iu}^H={\overline{X}}_i^T\widehat{\alpha }+{\widehat{u}}_i$ is the EBLUP estimator of ${\mu }_{iu}={\overline{X}}_i^T\alpha +u_i$ under the sample model (1.3) and $\widehat{b}$ is an estimator of $b$ in the model (1.2) for the weights $w_{j|i}.$ Note that $\left(\widehat{\alpha },{\widehat{u}}_i,{\widehat{\sigma }}_u^2,{\widehat{\sigma }}_h^2\right)$ is identical to $\left(\widehat{\beta },{\widehat{v}}_i,{\widehat{\sigma }}_v^2,{\widehat{\sigma }}_e^2\right)$ obtained by assuming that the population model (1.1) holds for the sample. Therefore, we can also express ${\widehat{\overline{Y}}}_i^{\text{PS}}$ as

$${\widehat{\overline{Y}}}_i^{\text{PS}}={\widehat{\overline{Y}}}_i^H+\left(1-\frac{n_i}{N_i}\right)\widehat{b}{\widehat{\sigma }}_e^2,(2.7)$$

noting that ${\widehat{\mu }}_i^H={\widehat{\mu }}_{iu}^H.$

The last term in (2.7) corrects for any bias due to informative sampling under (1.2). PS obtained the estimator $\widehat{b}$ of $b$ in (2.6) by regressing the sampling weights $w_{j|i}$ on $k_i\exp \left(x_{ij}^{T}a+b{y}_{ij}\right).$ The coefficients $k_i,a$ and $b$ may be estimated by fitting the model (1.2) using procedure NLIN in SAS or function nls in Splus. This involves iterative calculations and the initial values for $a$ and $b$ are obtained by regressing $\log \left(w_{j|i}\right)$ on $x_{ij}$ and $y_{ij}.$ Initial values for $k_i,i=1,\mathrm{...},M$ are taken as $k_i=N_i/n_i.$

2.2 MSE estimation

The mean squared error (MSE) of the EBLUP estimator ${\widehat{\mu }}_i^H,$ assuming non-informative sampling, is estimated by

$$\text{mse}\left({\widehat{\mu }}_i^H\right)=g_{1i}\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right)+g_{2i}\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right)+2g_{3i}\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right),(2.8)$$

where

$$\begin{array}{lll}{g}_{1i}\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right)\hfill & =\hfill & \left(1-{\widehat{\gamma }}_i\right){\widehat{\sigma }}_v^2,g_{2i}\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right)={\left({\overline{X}}_i-{\widehat{\gamma }}_i{\overline{x}}_i\right)}^T{\left({\displaystyle \sum _{i=1}^M{X}_i^T{\widehat{V}}_i^{-1}{X}_i}\right)}^{-1}\left({\overline{X}}_i-{\widehat{\gamma }}_i{\overline{x}}_i\right),\hfill \\ g_{3i}\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right)\hfill & =\hfill & {\widehat{\gamma }}_i{\left(1-{\widehat{\gamma }}_i\right)}^2{\widehat{\sigma }}_e^{-4}{\widehat{\sigma }}_v^{-2}h\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right),\hfill \\ h\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right)\hfill & =\hfill & {\widehat{\sigma }}_e^4\mathrm{var}\left({\widehat{\sigma }}_v^2\right)-2{\widehat{\sigma }}_e^2{\widehat{\sigma }}_v^2\mathrm{cov}\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right)+{\widehat{\sigma }}_v^4\mathrm{var}\left({\widehat{\sigma }}_e^2\right),\hfill \end{array}$$

${\widehat{V}}_i={\widehat{\sigma }}_e^2{I}_{n_i}+{\widehat{\sigma }}_v^2{1}_{n_i}{1}_{n_i}^T$ and $X_i^T=\left(x_{i1},\dots ,x_{i{n}_i}\right).$ The matrix ${\displaystyle {\sum }_{i=1}^M{X}_i^T{\widehat{V}}_i^{-1}{X}_i}$ may be expressed explicitly as ${\widehat{\sigma }}_e^{-2}{\displaystyle {\sum }_{i=1}^M{\displaystyle {\sum }_{j=1}^{n_i}\left(x_{ij}{x}_{ij}^T-{\widehat{\gamma }}_i{\overline{x}}_i{\overline{x}}_i^T\right)}}.$ The MSE estimator (2.8) is unbiased to second order under non-informative sampling (Rao 2003, Chapter 7). We refer the reader to Rao (2003, page 142) for the corresponding MSE estimator of ${\widehat{\overline{Y}}}_i^H.$

The MSE of the pseudo-EBLUP estimator ${\widehat{\mu }}_i^{\text{YR}}$ is estimated by

$$\text{mse}\left({\widehat{\mu }}_i^{\text{YR}}\right)=g_{1iw}\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right)+g_{2iw}\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right)+2g_{3iw}\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right),(2.9)$$

where

$$g_{1iw}\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right)=\left(1-{\widehat{\gamma }}_{iw}\right){\widehat{\sigma }}_v^2,g_{2iw}\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right)={\left({\overline{X}}_i-{\widehat{\gamma }}_{iw}{\overline{x}}_{iw}\right)}^T{\Phi }_w\left({\overline{X}}_i-{\widehat{\gamma }}_{iw}{\overline{x}}_{iw}\right),$$

$$\begin{array}{lll}{\Phi }_w\hfill & =\hfill & {\widehat{\sigma }}_e^2{\left({\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^{n_i}{x}_{ij}{z}_{ij}^T}}\right)}^{-1}\left({\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^{n_i}{z}_{ij}{z}_{ij}^T}}\right){\left\{{\left({\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^{n_i}{x}_{ij}{z}_{ij}^T}}\right)}^{-1}\right\}}^T\hfill \\ \hfill & +\hfill & {\widehat{\sigma }}_v^2{\left({\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^{n_i}{x}_{ij}{z}_{ij}^T}}\right)}^{-1}\left\{{\displaystyle \sum _{i=1}^M\left({\displaystyle \sum _{j=1}^{n_i}{z}_{ij}}\right){\left({\displaystyle \sum _{j=1}^{n_i}{z}_{ij}}\right)}^T}\right\}{\left\{{\left({\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^{n_i}{x}_{ij}{z}_{ij}^T}}\right)}^{-1}\right\}}^T,\hfill \\ \hfill & \hfill & g_{3iw}\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right)={\widehat{\gamma }}_{iw}{\left(1-{\widehat{\gamma }}_{iw}\right)}^2{\widehat{\sigma }}_e^{-4}{\widehat{\sigma }}_v^{-2}h\left({\widehat{\sigma }}_e^2,{\widehat{\sigma }}_v^2\right)\hfill \end{array}$$

and $z_{ij}=w_{ij}\left(x_{ij}-{\widehat{\gamma }}_{iw}{\overline{x}}_{iw}\right);$ see You and Rao (2002). The MSE estimator (2.9) is obtained by ignoring a cross-product term in $\text{MSE}\left({\widehat{\mu }}_i^{\text{YR}}\right).$ Torabi and Rao (2010) obtained a MSE estimator that accounts for the missing cross-product term and that is unbiased to second order under non-informative sampling. However, it is computationally more intensive than (2.9). It was not used in the simulation study (Section 4) since it would have slowed down the simulations significantly. A few simulation trials, however, revealed that the two MSE estimators give similar results under the simulation set-up used in Section 4.

Pfeffermann and Sverchkov (2007) proposed a parametric bootstrap method to estimate the MSE of the bias-adjusted estimator ${\widehat{\overline{Y}}}_i^{\text{PS}}$ given by (2.6). We have not included this MSE estimator in our simulation study.

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