Small area estimation using Fay-Herriot area level model with sampling variance smoothing and modeling
Section 5. Conclusion

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In this paper, we compare the model-based estimates under the Fay-Herriot model when sampling variances are smoothed and modeled. As in Hidiroglou et al. (2019), our results indicate that the Fay-Herriot model can provide great improvement for the direct survey estimates for LFS rate estimation, even though more complex models such as unmatched models or time series models could be used (e.g., You, 2008). Among all the estimators, FH-EBLUP and FH-HB using smoothed sampling variances perform the best in terms of ARE and CV reduction. Both FH-EBLUP and FH-HB using direct sampling variance estimates perform the worst. For HB modeling approach, both YLLM and STKM perform very well and are better than YCM, and YLLM is slightly better than STKM in our study. Thus if direct sampling variance estimates are used, YLLM or STKM model is suggested. Alternatively, smoothed sampling variances should be used in the Fay-Herriot model to overcome the sampling variance modeling difficulty as discussed in Section 3. The smoothed sampling variances based on the GVF model given by (2.2) in Section 2 can perform very well as shown in our study.

Appendix

Full conditional distributions and sampling procedure for YLLM

• $\left[{\theta }_i|y,\beta ,{\sigma }_i^2,{\sigma }_v^2\right]~N\left({\gamma }_i{y}_i+(1-{\gamma }_i){x_i^{\prime }}_{}\beta ,{\gamma }_i{\sigma }_i^2\right),$ where ${\gamma }_i={\sigma }_v^2/{\sigma }_v^2+{\sigma }_i^2,$ $i=1,\dots ,m;$
• $\left[\beta |y,\theta ,{\sigma }_i^2,{\sigma }_v^2\right]~N_p\left({\left({\displaystyle {\sum }_{i=1}^m{x}_i{x_i^{\prime }}_{}}\right)}^{-1}\left({\displaystyle {\sum }_{i=1}^m{x}_i{\theta }_i}\right),{\sigma }_v^2{\left({\displaystyle {\sum }_{i=1}^m{x}_i{x_i^{\prime }}_{}}\right)}^{-1}\right);$
• $\left[{\sigma }_v^2|y,\theta ,\beta ,{\sigma }_i^2\right]~\text{IG}\left(\frac{m}{2}-1,\frac{1}{2}{\displaystyle {\sum }_{i=1}^m{({\theta }_i-{x_i^{\prime }}_{}\beta )}^2}\right);$
• $\left[{\sigma }_i^2|y,\theta ,\beta ,{\sigma }_v^2,\delta ,{\tau }^2\right]\propto f({\sigma }_i^2)\cdot h({\sigma }_i^2),$ where $f({\sigma }_i^2)$ and $h({\sigma }_i^2)$ are $f({\sigma }_i^2)~\text{IG}\left(\frac{d_i+1}{2},\frac{{(y_i-{\theta }_i)}^2+d_i{s}_i^2}{2}\right),$ and $h({\sigma }_i^2)=\exp \left(-\frac{{(\log ({\sigma }_i^2)-{z_i^{\prime }}_{}\delta )}^2}{2{\tau }^2}\right);$
• $\left[\delta |y,\theta ,\beta ,{\sigma }_i^2,{\sigma }_v^2,{\tau }^2\right]~N_2\left({\left({\displaystyle {\sum }_{i=1}^m{z}_i{z_i^{\prime }}_{}}\right)}^{-1}\left({\displaystyle {\sum }_{i=1}^m{z}_i\log ({\sigma }_i^2)}\right),{\tau }^2{\left({\displaystyle {\sum }_{i=1}^m{z}_i{z_i^{\prime }}_{}}\right)}^{-1}\right);$
• $\left[{\tau }^2|y,\theta ,\beta ,{\sigma }_i^2,{\sigma }_v^2,\delta \right]~\text{IG}\left(\frac{m}{2}-1,\frac{1}{2}{\displaystyle {\sum }_{i=1}^m{\left(\log ({\sigma }_i^2)-{z_i^{\prime }}_{}\delta \right)}^2}\right).$

We use Metropolis-Hastings rejection step to update ${\sigma }_i^2$:

1. Draw ${\sigma }_i^{2*}$ from $\text{IG}\left(\frac{d_i+1}{2},\frac{{(y_i-{\theta }_i)}^2+d_i{s}_i^2}{2}\right);$
2. Compute the acceptance probability $\alpha ({\sigma }_i^{2*},{\sigma }_i^2{}^{(k)})=\min \left\{h({\sigma }_i^{2*})/h({\sigma }_i^2{}^{(k)}),1\right\};$
3. Generate $u$ from Uniform (0, 1), if $u<\alpha ({\sigma }_i^{2*},{\sigma }_i^2{}^{(k)}),$ the candidate ${\sigma }_i^{2*}$ is accepted, ${\sigma }_i^2{}^{(k+1)}={\sigma }_i^{2*};$ otherwise ${\sigma }_i^{2*}$ is rejected, and set ${\sigma }_i^2{}^{(k+1)}={\sigma }_i^2{}^{(k)}.$

Acknowledgements

I would like to thank the Editor, the Associate Editor and one referee for their constructive comments and suggestions to improve the paper.

References

Dass, S.C., Maiti, T., Ren, H. and Sinha, S. (2012). Confidence interval estimation of small area parameters shrinking both means and variances. Survey Methodology, 38, 2, 173-187. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2012002/article/11756-eng.pdf.

Fay, R.E., and Herriot, R.A. (1979). Estimates of income for small places: An application of James-Stein procedures to census data. Journal of the American Statistical Association, 74, 269-277.

Ghosh, M., Myung, J. and Moura, F.A.S. (2018). Robust Bayesian small area estimation. Survey Methodology, 44, 1, 101-115. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2018001/article/54959-eng.pdf.

Gilks, W.R., Richardson, S. and Spiegelhalter, D.J. (1996). Markov Chain Monte Carlo in Practice. Chapman & Hall/CRC.

Hidiroglou, M.A., Beaumont, J.-F. and Yung, W. (2019). Development of a small area estimation system at Statistics Canada. Survey Methodology, 45, 1, 101-126. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2019001/article/00009-eng.pdf.

Maples, J., Bell, W. and Huang, E. (2009). Small area variance modeling with application to county poverty estimates from the American community survey. Proceedings of the Section on Survey Research Methods, American Statistical Association, 5056-5067.

Molina, I, Nandram, B. and Rao, J.N.K. (2014). Small area estimation of general parameters with application to poverty indicators: A hierarchical Bayes approach. Annals of Applied Statistics, 8, 852-885.

Rao, J.N.K., and Molina, I. (2015). Small Area Estimation, 2^nd Edition. New York: John Wiley & Sons, Inc.

Rivest, L.P., and Vandal, N. (2002). Mean squared error estimation for small areas when the small area variances are estimated. Proceedings of the International Conference on Recent Advances in Survey Sampling, July 10-13, 2002, Ottawa, Canada.

Rubin-Bleuer, S., and You, Y. (2016). Comparison of some positive variance estimators for the Fay-Herriot small area model. Survey Methodology, 42, 1, 63-85. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2016001/article/14542-eng.pdf.

Souza, D.F., Moura, F.A.S. and Migon, H.S. (2009). Small area population prediction via hierarchical models. Survey Methodology, 35, 2, 203-214. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2009002/article/11042-eng.pdf.

Sugasawa, S., Tamae, H. and Kubokawa, T. (2017). Bayesian estimators for small area models shrinking both means and variances. Scandinavian Journal of Statistics, 44, 150-167.

Wang, J., and Fuller, W.A. (2003). The mean square error of small area predictors constructed with estimated area variances. Journal of the American Statistical Association, 98, 716-723.

You, Y. (2008). An integrated modeling approach to unemployment rate estimation for sub-provincial areas of Canada. Survey Methodology, 34, 1, 19-27. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2008001/article/10614-eng.pdf.

You, Y. (2010). Small area estimation under the Fay-Herriot model using different model variance estimation methods and different input sampling variances. Methodology branch working paper, SRID-2010-003E, Statistics Canada, Ottawa, Canada.

You, Y. (2016). Hierarchical Bayes sampling variance modeling for small area estimation based on area level models with applications. Methodology branch working paper, ICCSMD-2016-03-E, Statistics Canada, Ottawa, Canada.

You, Y., and Chapman, B. (2006). Small area estimation using area level models and estimated sampling variances. Survey Methodology, 32, 1, 97-103. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2006001/article/9263-eng.pdf.

You, Y., and Hidiroglou, M. (2012). Sampling variance smoothing methods for small area proportion estimators. Methodology Branch Working Paper, SRID-2012-08E, Statistics Canada, Ottawa, Canada; Presented as an invited paper at the Fields Institute Symposium on the Analysis of Survey Data and Small Area Estimation, Carleton University, Ottawa, 2012.

You, Y., and Rao, J.N.K. (2000). Hierarchical Bayes estimation of small area means using multi-level models. Survey Methodology, 26, 2, 173-181. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2000002/article/5537-eng.pdf.

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