Local polynomial estimation for a small area mean under informative sampling
Section 2. Existing methods

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

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

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 ${\widehat{v}}_i={\widehat{\gamma }}_i\left({\overline{y}}_i-{\overline{x}}_i^T\widehat{\beta }\right).$ The estimator of the regression vector $\beta$ in (1.1) is

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

The estimated variance components $\left({\widehat{\sigma }}_v^2,{\widehat{\sigma }}_e^2\right)$ are obtained by the Henderson method of fitting of constants (HFC) or restricted maximum likelihood (REML) (see Battese et al., 1988 and Chapter 7 in Rao and Molina, 2015). The EBLUP estimator of the area mean ${\overline{Y}}_i$ may be written in terms of ${\widehat{\mu }}_i^{\text{EBLUP}}$ as

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

Note that ${\widehat{\overline{Y}}}_i^{\text{EBLUP}}\approx {\widehat{\mu }}_i^{\text{EBLUP}}$ if the sampling fraction $n_i/N_i$ is sufficiently small. The EBLUP estimator ${\widehat{\overline{Y}}}_i^{\text{EBLUP}}$ is design consistent under simple random sampling (SRS) or stratified SRS with proportional allocation within small area $U_i,$ leading to equal $p_{j|i}’\text{s}\text{.}$

Pfeffermann and Sverchkov (2007) studied the estimation of small area means under informative sampling, assuming the following model for the sample data

$$y_{ij}=x_{ij}^T\alpha +u_i+h_{ij};j=1,\dots ,n_i;i=1,\dots ,M,(2.4)$$

where $u_i\stackrel{\text{iid}}{{\displaystyle \sim }}N\left(0,{\sigma }_u^2\right),$ and $h_{ij}|j\in s_i\stackrel{\text{iid}}{{\displaystyle \sim }}N\left(0,{\sigma }_h^2\right).$ They assumed that the unit design weight $w_{j|i}={\pi }_{j|i}^{-1}$ is random with conditional expectation

$$\begin{array}{ll}{E}_{si}\left(w_{j|i}|x_{ij},y_{ij},v_i\right)\hfill & =E_{si}\left(w_{j|i}|x_{ij},y_{ij}\right)\hfill \\ \hfill & =k_i\exp \left(x_{ij}^{T}a+d{y}_{ij}\right),(2.5)\hfill \end{array}$$

where $a$ and $d$ are fixed unknown constants and

$$k_i=\frac{N_i}{n_i}\left\{{\displaystyle \sum _{j=1}^{N_i}\exp \left(-x_{ij}^{T}a-d{y}_{ij}\right)/N_i}\right\}.$$

The Pfeffermann and Sverchkov (2007) estimator of ${\overline{Y}}_i$ provides protection against informative sampling supposing that this assumption holds. The estimator is given by

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

where ${\widehat{\mu }}_{iu}^{\text{EBLUP}}={\overline{X}}_i^T\widehat{\alpha }+{\widehat{u}}_i$ is the EBLUP estimator of ${\mu }_{iu}={\overline{X}}_i^T\alpha +u_i$ under the sample model (2.4) and $\widehat{d}$ is an estimator of $d$ in the model (2.5) for the weights $w_{j|i}.$ The last term in (2.6) corrects for any bias due to informative sampling under (2.5). Pfeffermann and Sverchkov (2007) obtained the estimator $\widehat{d}$ of $d$ in (2.5) by regressing the sampling weights $w_{j|i}$ on $k_i\exp \left(x_{ij}^{T}a+d{y}_{ij}\right).$ The coefficients $k_i,$ $a$ and $d$ may be estimated by fitting the model (2.5) using the NLIN procedure in SAS or function nls in Splus. This involves iterative calculations and the initial values for $a$ and $d$ are obtained by regressing $\log \left(w_{j|i}\right)$ on $x_{ij}$ and $y_{ij}.$ Initial values for ${\widehat{k}}_i,$ $i=1,\dots ,M$ are taken as $k_i=N_i/n_i.$

The Verret et al. (2015) estimator is obtained when the EBLUP theory is applied to model (1.2). Let $x_{ij}^{\text{aug}}={\left(x_{ij}^T,g\left(p_{j|i}\right)\right)}^T$ be the vector $x_{ij}$ augmented by the variable $g\left(p_{j|i}\right),$ ${\overline{G}}_i$ the area mean of the population values $g\left(p_{j|i}\right),$ and ${\mu }_{0i}={\overline{X}}_i^T{\beta }_0+{\overline{G}}_i{\delta }_0+v_{0i}.$ The EBLUP estimator of ${\mu }_{0i}$ is given by

$${\widehat{\mu }}_{0i}^{\text{EBLUP}}={\overline{X}}_i^T{\widehat{\beta }}_0+{\overline{G}}_i{\widehat{\delta }}_0+{\widehat{v}}_{0i}={\widehat{\gamma }}_{0i}{\overline{y}}_i+{\left({\overline{X}}_i-{\widehat{\gamma }}_{0i}{\overline{x}}_i\right)}^T{\widehat{\beta }}_0+\left({\overline{G}}_i-{\widehat{\gamma }}_{0i}{\overline{g}}_i\right){\widehat{\delta }}_0,(2.7)$$

where ${\widehat{\gamma }}_{0i}={\widehat{\sigma }}_{0v}^2/\left({\widehat{\sigma }}_{0v}^2+{\widehat{\sigma }}_{0e}^2/n_i\right),$ ${\overline{g}}_i={\displaystyle {\sum }_{j=1}^{n_i}g\left(p_{j|i}\right)/n_i}$ and ${\widehat{v}}_{0i}={\widehat{\gamma }}_{0i}\left({\overline{y}}_i-{\overline{x}}_i^T{\widehat{\beta }}_0-{\overline{g}}_i{\widehat{\delta }}_0\right).$ The parameters, $\left({\beta }_0,{\delta }_0\right)$ are estimated by

$${\left({\widehat{\beta }}_0^T,{\widehat{\delta }}_0\right)}^T={\left\{{\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^{n_i}{x}_{ij}^{\text{aug}}{\left(x_{ij}^{\text{aug}}-{\widehat{\gamma }}_{0i}{\overline{x}}_i^{\text{aug}}\right)}^T}}\right\}}^{-1}\left\{{\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^{n_i}\left(x_{ij}^{\text{aug}}-{\widehat{\gamma }}_{0i}{\overline{x}}_i^{\text{aug}}\right)}}{y}_{ij}\right\},(2.8)$$

with ${\overline{x}}_i^{\text{aug}}={\displaystyle {\sum }_{j=1}^{n_i}{x}_{ij}^{\text{aug}}}/n_i={\left({\overline{x}}_i^T,{\overline{g}}_i\right)}^T.$ The model parameters $\left({\widehat{\sigma }}_{0v}^2,{\widehat{\sigma }}_{0e}^2\right)$ are estimated by HFC or REML method. The estimator of the area mean ${\overline{Y}}_i,$ denoted ${\widehat{\overline{Y}}}_i^{\text{VRH}},$ may be written in terms of ${\widehat{\mu }}_{0i}^{\text{EBLUP}}$ as

$${\widehat{\overline{Y}}}_i^{\text{VRH}}=\frac{1}{N_i}\left[\left(N_i-n_i\right){\widehat{\mu }}_{0i}^{\text{EBLUP}}+n_i\left\{{\overline{y}}_i+{\left({\overline{X}}_i-{\overline{x}}_i\right)}^T{\widehat{\beta }}_0+{\left({\overline{G}}_i-{\overline{g}}_i\right)}^T{\widehat{\delta }}_0\right\}\right].(2.9)$$

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