Small area estimation using Fay-Herriot area level model with sampling variance smoothing and modeling
Section 1. Introduction

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Small area estimation is popular and important in survey data analysis. Model-based estimates have been widely used in practice to provide reliable estimates for small areas. In practice, area level models are usually used whenever direct survey estimates and area level auxiliary variables are available. Various area level models have been proposed to improve the precision of the direct survey estimates, see Rao and Molina (2015). Among the area level models, the Fay-Herriot model (Fay and Herriot, 1979) is a basic area level model widely used in small area estimation. The Fay-Herriot model has two components, namely, a sampling model for the direct survey estimates and a linking model for the small area parameter of interest. The sampling model assumes that a direct survey estimator $y_i$ is design unbiased for the small area parameter ${\theta }_i$ such that

$$y_i={\theta }_i+e_i,i=1,\dots ,m,(1.1)$$

where $e_i$ is the sampling error associated with the direct estimator $y_i$ and $m$ is the number of small areas. It is customary to assume that $e_i’\text{s}$ are independently normal random variables with mean $\text{E}(e_i)=0$ and sampling variance $Var(e_i)={\sigma }_i^2.$ The linking model assumes that the small area parameter ${\theta }_i$ is related to auxiliary variables $x_i=(x_{i1},\dots ,x_{ip}{)}^{\prime }$ through a linear regression model given as

$${\theta }_i={x_i^{\prime }}_{}\beta +v_i,i=1,\dots ,m,(1.2)$$

where $\beta =({\beta }_1,\dots ,{\beta }_p{)}^{\prime }$ is a $p\times 1$ vector of regression coefficients, and the $v_i’\text{s}$ are area-specific random effects assumed to be independent and identically distributed with $\text{E}(v_i)=0$ and $Var(v_i)={\sigma }_v^2.$ The assumption of normality is generally included. Random effects $v_i$ and sampling errors $e_i$ are mutually independent. The model variance ${\sigma }_v^2$ is unknown and needs to be estimated. Combining models (1.1) and (1.2) leads to a linear mixed model given as

$$y_i={x_i^{\prime }}_{}\beta +v_i+e_i,i=1,\dots ,m.(1.3)$$

Model (1.3) involves both design-based random errors $e_i$ and model-based random effects $v_i.$ For the Fay-Herriot model, the sampling variance ${\sigma }_i^2$ is usually assumed to be known. This is a very strong assumption. In practice, unbiased direct estimates of the sampling variances are generally available. To make use of the direct sampling variance estimates, two approaches are available in practice, namely, smoothing and modeling. For the smoothing approach, smoothed estimates of the sampling variances are used in the Fay-Herriot model and then treated as known. The smoothing approach requires external variables and external models such as use of the generalized variance function (GVF) and design effects. You and Hidiroglou (2012) particularly studied the GVF and design effects methods for sampling variance smoothing for proportions. In this paper, we will use a GVF model proposed in You and Hidiroglou (2012) for the sampling variance smoothing.

As an alternative to smoothing, sampling variance modeling is also commonly used in practice. Let $s_i^2$ denote the direct estimator for the sampling variance ${\sigma }_i^2.$ We consider a custom model for $s_i^2$ as $d_i{s}_i^2~{\sigma }_i^2{\chi }_{d_i}^2,$ where $d_i=n_i-1$ and $n_i$ is the sample size for the $i^{\text{th}}$ area. Rivest and Vandal (2002) and Wang and Fuller (2003) used empirical best linear unbiased prediction (EBLUP) method to obtain the model-based estimates. You and Chapman (2006) considered a hierarchical Bayes (HB) approach and combined the sampling variance model $d_i{s}_i^2~{\sigma }_i^2{\chi }_{d_i}^2$ with the small area model (1.3) to construct an integrated model. The integrated model borrows strength for small area estimates and sampling variance estimates simultaneously. The integrated HB modeling approach with $d_i{s}_i^2~{\sigma }_i^2{\chi }_{d_i}^2$ has thus been widely used in practice, for example, You (2008, 2016), Dass, Maiti, Ren and Sinha (2012), Sugasawa, Tamae and Kubokawa (2017), Ghosh, Myung and Moura (2018), and Hidiroglou, Beaumont and Yung (2019).

In this paper, we consider both the smoothing and modeling approaches for the sampling variances. In Section 2, we present the EBLUP method based on both the smoothed and direct estimates of the sampling variances. In Section 3, we present the Fay-Herriot HB model and three other HB models based on sampling variance modeling. We compare the effects of sampling variance smoothing and modeling in Section 4 through a real data analysis, and we offer some suggestions in Section 5.

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