Small area estimation using Fay-Herriot area level model with sampling variance smoothing and modeling
Section 3. Fay-Herriot model using HB approach with sampling variance modeling

• Previous
• Next

In this section we first present the Fay-Herriot model in a HB framework. Then we consider three models for the sampling variance modeling. The first model is the one considered in You and Chapman (2006) in which an inverse gamma model is used for the sampling variance ${\sigma }_i^2$ with known vague parameter values. The second model is introduced in You (2016) whereby a log-linear model with random error is used for ${\sigma }_i^2.$ The third model is one proposed by Sugasawa et al. (2017) where an inverse gamma model is used for ${\sigma }_i^2$ but with different parameter settings.

HB Model 1: Fay-Herriot model in HB, denoted as FH-HB:

• $y_i|{\theta }_i,{\sigma }_i^2~\text{ind}N({\theta }_i,{\sigma }_i^2),i=1,\dots ,m;$
• ${\theta }_i|\beta ,{\sigma }_v^2~\text{ind}N({x_i^{\prime }}_{}\beta ,{\sigma }_v^2),i=1,\dots ,m;$
• Flat priors for unknown parameters: $\pi (\beta )\propto 1,$ $\pi ({\sigma }_v^2)\propto 1.$

Note that in the FH-HB model, the sampling variance ${\sigma }_i^2$ is assumed to be known. Either a smoothed sampling variance ${\tilde{\sigma }}_i^2$ or a direct sampling variance estimate $s_i^2$ will be used in place of ${\sigma }_i^2.$

HB Model 2: You-Chapman Model (You and Chapman, 2006), denoted as YCM:

• $y_i|{\theta }_i,{\sigma }_i^2~\text{ind}N({\theta }_i,{\sigma }_i^2),i=1,\dots ,m;$
• $d_i{s}_i^2|{\sigma }_i^2~\text{ind}{\sigma }_i^2{\chi }_{d_i}^2,$ $d_i=n_i-1,i=1,\dots ,m;$
• ${\theta }_i|\beta ,{\sigma }_v^2~\text{ind}N({x_i^{\prime }}_{}\beta ,{\sigma }_v^2),i=1,\dots ,m;$
• $\pi ({\sigma }_i^2)~\text{IG}(a_i,b_i),$ where $a_i=$ 0.0001, $b_i=$ 0.0001, $i=1,\dots ,m;$
• Flat priors for unknown parameters: $\pi (\beta )\propto 1,$ $\pi ({\sigma }_v^2)\propto 1.$

The full conditional distributions for the Gibbs sampling procedure under both FH-HB and YCM can be found in You and Chapman (2006).

HB Model 3: You (2016) Log-linear model on sampling variances, denoted as YLLM:

• $y_i|{\theta }_i,{\sigma }_i^2~\text{ind}N({\theta }_i,{\sigma }_i^2),i=1,\dots ,m;$
• $d_i{s}_i^2|{\sigma }_i^2~\text{ind}{\sigma }_i^2{\chi }_{d_i}^2,d_i=n_i-1,i=1,\dots ,m;$
• ${\theta }_i|\beta ,{\sigma }_v^2~\text{ind}N({x_i^{\prime }}_{}\beta ,{\sigma }_v^2),i=1,\dots ,m;$
• $\log ({\sigma }_i^2)~N\left({\delta }_1+{\delta }_2\log (n_i),{\tau }^2\right),i=1,\dots ,m;$
• Flat priors for unknown parameters: $\pi (\beta )\propto 1,$ $\pi ({\delta }_1,{\delta }_2)\propto 1,$ $\pi ({\sigma }_v^2)\propto 1,$ $\pi ({\tau }^2)\propto 1.$

Note that model YLLM uses a log-linear model for the sampling variance ${\sigma }_i^2,$ and extends the model proposed by Souza, Moura and Migon (2009) for sampling variances by using $\log (n_i)$ and adding a random effect to the regression part in the model. The full conditional distributions for the Gibbs sampling procedure are given in the Appendix.

HB Model 4: Sugasawa, Tamae and Kubokawa (2017) model shrinking both means and variances, denoted as STKM:

• $y_i|{\theta }_i,{\sigma }_i^2~\text{ind}N({\theta }_i,{\sigma }_i^2),i=1,\dots ,m;$
• $d_i{s}_i^2|{\sigma }_i^2~\text{ind}{\sigma }_i^2{\chi }_{d_i}^2,d_i=n_i-1,i=1,\dots ,m;$
• ${\theta }_i|\beta ,{\sigma }_v^2~\text{ind}N({x_i^{\prime }}_{}\beta ,{\sigma }_v^2),i=1,\dots ,m;$
• $\pi ({\sigma }_i^2)~\text{IG}(a_i,b_i\gamma ),$ where $a_i$ and $b_i$ are known constants, $a_i=O(1),$ $b_i=O(n_i^{-1});$
• Flat priors for unknown parameters: $\pi (\beta )\propto 1,$ $\pi ({\sigma }_v^2)\propto 1,$ $\pi (\gamma )\propto 1.$

Note that in STKM, for the inverse gamma model of ${\sigma }_i^2,$ we choose $a_i=2$ and $b_i=n_i^{-1}$ as suggested by Sugasawa et al. (2017). Ghosh et al. (2018) also used the same setting in their study of comparing HB estimators. The full conditional distributions for STKM can be found in Sugasawa et al. (2017).

Note that the Chi-squared sampling variance modeling $d_i{s}_i^2~{\sigma }_i^2{\chi }_{d_i}^2$ in the above HB Models 2-4 is based on normality and simple random sampling (Rivest and Vandal, 2002). For complex survey designs, the degrees of freedom $d_i$ may need to be determined more carefully. There is no sound theoretical result for determining the degrees of freedom (Dass et al., 2012). The approximation formula based on non-normal unit level errors provided by Wang and Fuller (2003) and the simulation based guideline of Maples, Bell and Huang (2009) could be useful but require unit level data and an extensive simulation study. A careful determination of the degrees of freedom may provide a reasonably useful approximation. Moreover, Bayesian model fit analysis can also be helpful for model determination.

• Previous
• Article main page
• Next