Model-based small area estimation under informative sampling 1. IntroductionModel-based small area estimation under informative sampling 1. Introduction

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Estimates of population totals and means are often required for small subpopulations (or areas). Traditional area-specific direct estimators are not reliable if the area sample size is small. As a result, it becomes necessary to “borrow strength” across areas through indirect estimation based on models that provide a link to related areas. Linking models make use of auxiliary population information either at the area level or at the unit level. Rao (2003, Chapter 7) gives a detailed account of area level and unit level models that are widely used for small area estimation.

Suppose that the population of interest, $U,$ consists of $M$ non-overlapping areas with $N_i$ elements in the $i^{\text{th}}$ area $\left(i=1,\dots ,M\right).$ A sample, $s,$ of $m$ areas is first selected using a specified sampling scheme with inclusion probabilities ${\pi }_i=m{p}_i\left(i=1,\dots ,M\right),$ where $p_i$ denotes the selection probability of area $i.$ Subsamples $s_i$ of specified sizes $n_i$ are then independently selected from the sampled areas $i$ according to specified sampling schemes with selection probabilities $p_{j|i}\left({\displaystyle {\sum }_{j=1}^{N_i}{p}_{j|i}}=1\right)$ such that the second-stage inclusion probabilities are ${\pi }_{j|i}=n_i{p}_{j|i}$ for unit $j$ in area $i$ $\left(j=1,\dots ,N_i\right).$ Typically, the selection probability $p_{j|i}=b_{ij}/{\displaystyle {\sum }_{k=1}^{N_i}{b}_{ik}},$ where $b_{ij}$ is a size measure related to the response variable $y_{ij}.$ In this paper, we focus on the special case where all the areas are sampled, $m=M.$

We assume a nested error linear regression model for the population, based on covariates $x_{ij}$ related to the response variable $y_{ij}.$ The population model is assumed to be given by

$$y_{ij}=x_{ij}^T\beta +v_i+e_{ij};j=1,\dots ,N_i;i=1,\dots ,M,(1.1)$$

where $v_i\stackrel{\text{iid}}{\sim }N\left(0,{\sigma }_v^2\right)$ are random small area effects that are independent of the unit-level errors $e_{ij}\stackrel{\text{iid}}{\sim }N\left(0,{\sigma }_e^2\right),x_{ij}={\left(1,x_{ij1},x_{ij2},\dots ,x_{ijp}\right)}^T$ and $\beta ={\left({\beta }_0,{\beta }_1,\dots ,{\beta }_p\right)}^T.$ Parameters of interest are the small area means ${\overline{Y}}_i=N_i^{-1}{\displaystyle {\sum }_{j=1}^{N_i}{y}_{ij}}$ which may be approximated by ${\mu }_i={\overline{X}}_i^T\beta +v_i,$ if the area sizes $N_i$ are large, where ${\overline{X}}_i=N_i^{-1}{\displaystyle {\sum }_{j=1}^{N_i}{x}_{ij}}$ is the known population mean of $x$ for area $i.$

Efficient model-based estimators of the area means ${\mu }_i$ may be obtained if the sampling design is non-informative for the model, which implies that the sample and the population models coincide. In particular, empirical best linear unbiased prediction (EBLUP) estimators (Henderson 1975), based on the assumed sample model under non-informative sampling, may be used to estimate small area means ${\overline{Y}}_i$ or ${\mu }_i$ (see Section 2 and Rao 2003, Chapter 7). However, in many practical situations the selection probabilities $p_{j|i}$ may be related to the associated $y_{ij}$ even after conditioning on the covariates $x_{ij}.$ In such cases, we have “informative sampling” in the sense that the population model (1.1) no longer holds for the sample. For example, Pfeffermann and Sverchkov (2007) assumed that the sampled unit design weight $w_{j|i}={\pi }_{j|i}^{-1}$ is random with conditional expectation

$$\begin{array}{lll}{E}_{s_i}\left(w_{j|i}|x_{ij},y_{ij},v_i\right)\hfill & =\hfill & E_{s_i}\left(w_{j|i}|x_{ij},y_{ij}\right)\hfill \\ \hfill & =\hfill & k_i\exp \left(x_{ij}^{T}a+b{y}_{ij}\right),\hfill \end{array}(1.2)$$

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

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

Under informative sampling within areas, the EBLUP estimator of ${\overline{Y}}_i,$ assuming that model (1.1) holds for the sample, may be heavily biased. It is, therefore, necessary to develop estimators that can account for sample selection bias and thus reduce estimation bias. Pfeffermann and Sverchkov (2007) developed a bias-adjusted estimator of the mean ${\overline{Y}}_i$ under the assumption (1.2) on the design weights $w_{j|i}$ and assuming that the sample model is a nested error model

$$y_{ij}=x_{ij}^T\alpha +u_i+h_{ij};j=1,\dots ,n_i;i=1,\dots ,M,(1.3)$$

where $u_i\stackrel{\text{iid}}{\sim }N\left(0,{\sigma }_u^2\right)\text{,}$ and $h_{ij}|j\in s_i\stackrel{\text{iid}}{\sim }N\left(0,{\sigma }_h^2\right).$ Pfeffermann and Sverchkov (2007) noted that under a sampling scheme satisfying (1.2) the population model is also a nested error model but with different parameters. However, they do not use the form of the population model. The sample model (1.3) is identified after fitting the model to the sample data and then doing some model diagnostics. Similarly, model (1.2) on the weights is identified from the sample data $\left\{w_{j|i},y_{ij},x_{ij},j\in s_i\text{,}i\in s\right\}.$ Their estimators are noted (PS) in the following.

Prasad and Rao (1999) and You and Rao (2002) developed pseudo-EBLUP estimators of small area means ${\mu }_i$ that depend on the sampling weights $w_{j|i},$ assuming non-informative sampling for the model (1.1). Their motivation for pseudo-EBLUP is to ensure design consistency as the area sample size, $n_i,$ increases. The estimators of You and Rao (note (YR) in the following) also satisfy a benchmarking property in the sense that the associated estimators of area totals add up to a reliable direct estimator of the total, unlike the EBLUP estimators. Stefan (2005) studied the empirical performance of pseudo-EBLUP estimators under informative sampling for model (1.1) and showed that the pseudo-EBLUP leads to smaller bias compared to the EBLUP.

The main purpose of our paper is to study augmented sample models of the form

$$y_{ij}=x_{ij}^T{\beta }_0+g\left(p_{j|i}\right){\delta }_0+{\tilde{v}}_i+{\tilde{e}}_{ij};j=1,\dots ,n_i;i=1,\dots ,M(1.4)$$

for a suitably defined function $g_{ij}=g\left(p_{j|i}\right)$ of the probability $p_{j|i},$ where ${\tilde{v}}_i\stackrel{\text{iid}}{\sim }N\left(0,{\sigma }_{v0}^2\right)$ and independent of ${\tilde{e}}_{ij}\stackrel{\text{iid}}{\sim }N\left(0,{\sigma }_{e0}^2\right),$ and ${\beta }_0={\left({\beta }_{00},{\beta }_{01},\dots ,{\beta }_{0p}\right)}^T.$ The sample model (1.4) is identified after fitting the model to sample data for different choices of the function $g(\cdot )$ and checking their adequacy. For example, residuals $r_{ij}$ from fitting the model (1.4) without the augmenting variable $g\left(p_{j|i}\right)$ may be plotted against $g\left(p_{j|i}\right)$ to select $g(\cdot ).$ The identified augmented sample model will also hold for the population (Skinner 1994, Rao 2003, Section 5.3). Possible choices of $g\left(p_{j|i}\right)$ are $p_{j|i},\log p_{j|i},w_{j|i}$ and $n_i{w}_{j|i}=p_{j|i}^{-1}.$

From the augmented sample model (1.4) we obtain the EBLUP estimators of ${\overline{Y}}_i$ or ${\mu }_i={\overline{X}}_i^T{\beta }_0+{\overline{G}}_i{\delta }_0+{\tilde{v}}_i,$ the approximate area mean under the augmented population model, where ${\overline{G}}_i$ is the area mean of the population values $g\left(p_{j|i}\right)\equiv g_{ij}.$ The EBLUP of ${\overline{Y}}_i$ or ${\mu }_i$ requires the knowledge of ${\overline{G}}_i$ which depends on all the population values $p_{j|i}.$ However, the choice $g\left(p_{j|i}\right)=p_{j|i}$ gives ${\overline{G}}_i=1/N_i$ and the choice $g\left(p_{j|i}\right)=n_i{w}_{j|i}$ gives ${\overline{G}}_i=n_i{\overline{W}}_i,$ where ${\overline{W}}_i$ is the area population mean of the weights $w_{j|i}.$ The means ${\overline{W}}_i$ are often known in practice. Pseudo-EBLUP estimators under the augmented model are also studied.

We conducted a simulation study under the design-model (or pm) framework to study the bias and MSE of the proposed estimators relative to EBLUP and pseudo-EBLUP estimators based on non-informative sampling, and the bias-adjusted estimators of Pfeffermann and Sverchkov (2007). We also studied the performance of MSE estimators in terms of relative bias.

Section 2 summarizes the existing model-based methods for estimating the small area means ${\overline{Y}}_i$ or ${\mu }_i.$ Proposed methods based on the augmented sample model (1.4) are presented in Section 3. The results of the simulation study are reported in Section 4. Concluding remarks are given in Section 5.

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