Small area benchmarked estimation under the basic unit level model when the sampling rates are non‑negligible
Section 6. Conclusion

• Previous
• Next

In general, the sum of model-based small area estimates is not equal to a direct estimate obtained across the union of these small areas. The weight that is associated with the direct estimator can be the sampling weight or one obtained as a result of using the GREG estimator. The auxiliary data that are used to obtain the GREG and the unit-level small area estimates may not necessarily coincide. In this paper, we have suggested several benchmarking procedures for two well-known small area estimators (EBLUP and YR) that are based on the unit level model. We considered the case when the sampling rates are not negligible, and that the sample design is ignorable. In the event that it is deemed that the sample design is not ignorable for some of the survey items, the auxiliary data vector $x_{ij}$ in model (2.2) could be augmented by including an additional variable $g_{ij}$ specified function of the survey weights to offset the potential bias of the EBLUP or YR estimators. Verret et al. (2015) proposed a number of choices for $g_{ij}$ that included the survey weight $w_{ij}.$ In the case of the EBLUP estimator, benchmarking is achieved by adding the variable $q_{ij}=w_{ij}^{\text{GREG}}-1.$ Since $q_{ij}$ should be highly correlated to $w_{ij},$ the suggested procedure for benchmarking EBLUP should provide good protection against possible non ignorable sampling. The simulations in Verret et al. (2015) illustrated that the YR procedure, on its own, provides good protection as well against possible non ignorable sampling. Their simulation also showed that further protection can be obtained by their setting $g_{ij}$ equal to $n_i{w}_{ij}.$

We extended the benchmarking procedures in Stefan and Hidiroglou (2020) to the case of non‑negligible sampling rates within each small area. These procedures are based on estimators that were initially developed by Battese et al. (1988) (EBLUP estimator), and You and Rao (2002) (YR estimator) when the sampling rates within each small area are negligible. Ugarte et al. (2009) proposed a different benchmarked estimator which is a restricted EBLUP estimator. We extended the procedure in Ugarte et al. (2009) to obtain a benchmarked estimator that incorporates the survey weights, and that is essentially a restricted YR estimator. We also considered two benchmarked estimators based on simple ratio adjustments applied on the EBLUP and YR estimators respectively. We carried out a simulation study to compare the properties of these six benchmarked estimators.

If the auxiliary data used to estimate the small area means are the same as those used in the GREG, and if the model is correct, the restricted procedure in Ugarte et al. (2009) and the ratio adjusted EBLUP estimator will have the smaller $\overline{\text{ARB}}’s$ and $\overline{\text{RRMSE}}’s.$ On the other hand, if the model is incorrect and the auxiliary data are the same ones, the YR estimator based on Stefan and Hidiroglou (2020) procedure, adapted to non‑negligible sampling rates, has the smallest $\overline{\text{ARB}}’s,$ whereas the restricted YR estimator has the smallest $\overline{\text{RRMSE}}’s.$ On the other hand, if the auxiliary data used to estimate the small area means are not the same as those used in the GREG, we come to the following conclusions. The restricted EBLUP and the ratio adjusted EBLUP estimators are the benchmarked estimators that have the smallest $\overline{\text{ARB}}’s$ and $\overline{\text{RRMSE}}’s$ if the model is correct. If the model is not correct, the restricted YR estimator is the preferred choice both in terms of $\overline{\text{ARB}}$ and $\overline{\text{RRMSE}}.$

Benchmarking should be based on the EBLUP procedure if the linear mixed effects model is appropriate. If the linear model and the benchmark (the GREG estimator) have in common a large amount of auxiliary information, the benchmarked estimators are similar to their non benchmarked versions, otherwise the loss of efficiency due to benchmarking may be important. If the model is not correct, the YR procedure should be used to achieve benchmarking. In this case, benchmarking may bring about important gains in terms of $\overline{\text{ARB}}$ and $\overline{\text{RRMSE}},$ especially if the small area model and the GREG estimator share a small number of auxiliary variables. The finite populations associated with incorrect modeling were generated based on model (4.2), with mean function incorrectly specified. However, there are many ways in which a model may be wrong, and the conclusions associated with these cases may be different.

Acknowledgements

We would like to thank the two anonymous referees and the associate editor for their constructive suggestions.

Appendix A

Proof of Result 1. The EBLUP estimators ${\widehat{\beta }}_a={({({\widehat{\beta }}_{1a})}^T,{\widehat{\beta }}_{2a})}^T$ and ${\widehat{v}}_{ia},$ that are based on model (3.6), satisfy the equation

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}\left(\begin{array}{l}{x}_{ij}\\ q_{ij}\end{array}\right)\left(y_{ij}-x_{ij}^T{\widehat{\beta }}_{1a}-q_{ij}{\widehat{\beta }}_{2a}-{\widehat{v}}_{ia}\right)}}=0.(\text{A}.1)$$

Equation (A.1) has the form of equation (2.10) and corresponds to augmented model (3.6). Expanding the second equation in (A.1), we obtain that

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{q}_{ij}{x}_{ij}^T{\widehat{\beta }}_{1a}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{q}_{ij}^2{\widehat{\beta }}_{2a}}}+{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{q}_{ij}{\widehat{v}}_{ia}}}={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{q}_{ij}{y}_{ij}}}.(\text{A}.2)$$

The variable $q_{ij}$ is defined as $q_{ij}=w_{ij}^{\text{GREG}}-1.$ The right-hand side of (A.2) is

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{q}_{ij}{y}_{ij}}}={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}\left(w_{ij}^{\text{GREG}}-1\right)y_{ij}}}={\widehat{Y}}^{\text{GREG}}-{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{y}_{ij}}.}(\text{A}.3)$$

The sums that appear on the left-hand side of (A.2) are given by

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{q}_{ij}{x}_{ij}^T}}={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}\left(w_{ij}^{\text{GREG}}-1\right)x_{ij}^T}}={\left({\displaystyle \sum _{i=1}^m{X}_i}-{\displaystyle \sum _{j\in s_i}{x}_{ij}}\right)}^T={\left({\displaystyle \sum _{i=1}^m{x}_{ir}}\right)}^T,(\text{A}.4)$$

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{q}_{ij}^2}=}{\displaystyle \sum _{i=1}^m{q}_{iw}},(\text{A}.5)$$

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{q}_{ij}}}={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}\left(w_{ij}^{\text{GREG}}-1\right)}}={\displaystyle \sum _{i=1}^m\left({\widehat{N}}_i^{\text{GREG}}-n_i\right)}.(\text{A}.6)$$

In establishing that last equality of (A.4), we used that $x_{ij}\subseteq x_{ij}^{*}$ and that weights $w_{ij}^{\text{GREG}}$ satisfy equation (3.3). Result 1 follows by replacing (A.3), (A.4), (A.5) and (A.6) into (A.2).

Proof of Result 2. The survey-weighted estimating equations that defines ${\widehat{\beta }}^{\text{YR}}$ and ${\widehat{v}}^{\text{YR}}$ are given by (2.12) constructed with the weights $w_{ij}^{\text{GREG}}-1\text{:}$

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}\left(w_{ij}^{\text{GREG}}-1\right)x_{ij}\left(y_{ij}-x_{ij}^T{\widehat{\beta }}^{\text{YR}}-{\widehat{v}}_i^{\text{YR}}\right)}=0}\text{.}$$

Since the first term of $x_{ij}$ is one (representing an intercept), it follows that

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}\left(w_{ij}^{\text{GREG}}-1\right)\left(y_{ij}-x_{ij}^T{\widehat{\beta }}^{\text{YR}}-{\widehat{v}}_i^{\text{YR}}\right)}=0}.(\text{A}.7)$$

The terms in (A.7) are given by:

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}\left(w_{ij}^{\text{GREG}}-1\right)y_{ij}}={\widehat{Y}}^{\text{GREG}}-{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{y}_{ij}}}},(\text{A}.8)$$

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}\left(w_{ij}^{\text{GREG}}-1\right)x_{ij}^T{\widehat{\beta }}^{\text{YR}}}={\left({\displaystyle \sum _{i=1}^m{X}_i}-{\displaystyle \sum _{j\in s_i}{x}_{ij}}\right)}^T}{\widehat{\beta }}^{\text{YR}}={\left({\displaystyle \sum _{i=1}^m{x}_{ir}}\right)}^T{\widehat{\beta }}^{\text{YR}},(\text{A}.9)$$

and

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}\left(w_{ij}^{\text{GREG}}-1\right){\widehat{v}}_i^{\text{YR}}}={\displaystyle \sum _{i=1}^m\left({\widehat{N}}_i^{\text{GREG}}-n_i\right){\widehat{v}}_i^{\text{YR}}}}.(\text{A}.10)$$

Plugging (A.8), (A.9) and (A.10) into (A.7) leads to

$${\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j\in s_i}{y}_{ij}}}+{\left({\displaystyle \sum _{i=1}^m{x}_{ri}}\right)}^T{\widehat{\beta }}^{\text{YR}}+{\displaystyle \sum _{i=1}^m\left({\widehat{N}}_i^{\text{GREG}}-n_i\right){\widehat{v}}_i^{\text{YR}}}={\widehat{Y}}^{\text{GREG}}.(\text{A}.11)$$

Equation (A.11) proves Result 2.

Appendix B

Re-parameterized REML estimation of variance components

Let $\delta =\left({\delta }_1,{\delta }_2\right)$ be the vector of variance components, where ${\delta }_1={\sigma }_v^2$ and ${\delta }_2={\sigma }_e^2.$ We define the vector $\alpha =\left({\alpha }_1,{\alpha }_2\right)$ such that ${\sigma }_v^2=e^{{\alpha }_1}$ and ${\sigma }_e^2=e^{{\alpha }_2}.$ The restricted maximum log-likelihood function, denoted as $l\left(\text{}\alpha \text{}\right)$ is

$$l\left(\text{}\alpha \text{}\right)=l\left({\alpha }_1,{\alpha }_2\right)=c-\frac{1}{2}\log \left|V\right|-\frac{1}{2}\log \left|X^T{V}^{-1}X\right|-\frac{1}{2}{y}^{T}Py,(\text{B}.1)$$

where $c$ is a generic constant, $V=e^{{\alpha }_1}Z{Z}^T+e^{{\alpha }_2}{I}_n$ and $P=V^{-1}-V^{-1}X{(X^T{V}^{-1}X)}^{-1}{X}^T{V}^{-1}.$ Notice that $PX=0.$ The solution to the maximization of $l\left(\text{}\alpha \text{}\right)$ is obtained iteratively using the Fisher-scoring algorithm by updating the following equation

$${\alpha }^{(r+1)}={\alpha }^{(r)}+I{({\alpha }^{(r)})}^{-1}s({\alpha }^{(r)}).(\text{B}.2)$$

Here, $s({\alpha }^{(r)})=({\partial l({\alpha }^{(r)})/\partial {\alpha }_1,\partial l({\alpha }^{(r)})/\partial {\alpha }_2)}^T$ is the vector of first-order partial derivatives of $l\left(\text{}\alpha \text{}\right)$ with respect to $\alpha ,$ and $I\left({\alpha }^{(r)}\right)={(I_{jk}\left({\alpha }^{(r)}\right))}_{j,k=1,2}$ is the matrix of expected second-order derivatives of $-l\left(\text{}\alpha \text{}\right)$ with respect to $\alpha ,$ where $I_{jk}({\alpha }^{(r)})=E(-{\partial }^{2}l({\alpha }^{(r)})/\partial {\alpha }_j\partial {\alpha }_k).$

Under the BHF model, the first-order partial derivatives of $l\left(\text{}\alpha \text{}\right)$ are given by

$$\frac{\partial l}{\partial {\alpha }_j}(\alpha )=\left[-\frac{1}{2}\text{tr}\text{(}P{V}_{(j)}\text{)}+\frac{1}{2}{y}^{T}P{V}_{(j)}Py\right]e^{{\alpha }_j},j=1,2,(\text{B}.3)$$

where $V_{(1)}=Z{Z}^T$ and $V_{(2)}=I_n.$ The expected values of the second-order partial derivatives of $l\left(\text{}\alpha \text{}\right)$ are

$$E\left(\frac{{\partial }^{2}l}{\partial {\alpha }_j\partial {\alpha }_k}(\alpha )\right)=-\frac{1}{2}\text{tr}\text{(}P{V}_{(j)}P{V}_{(k)}\text{)}{e}^{{\alpha }_j+{\alpha }_k},j,k=1,2.(\text{B}.4)$$

The re-parameterized REML estimator of $\delta$ is obtained as

$${\widehat{\delta }}^{\text{reRE}}=\left({\widehat{\sigma }}_v^{2\text{reRE}},{\widehat{\sigma }}_e^{2\text{reRE}}\right)=\left(e^{{\widehat{\alpha }}_1},e^{{\widehat{\alpha }}_2}\right).(\text{B}.5)$$

References

Battese, G.E., Harter, R.M. and Fuller, W.A. (1988). An error component model for prediction of county crop areas using survey and satellite data. Journal of the American Statistical Association, 83, 28-36.

Bell, W.R., Datta, G.S. and Ghosh, M. (2013). Benchmarking small area estimators. Biometrika, 100, 189-202.

Datta, G.S., Ghosh, M., Steorts, R. and Maples, J. (2011). Bayesian benchmarking with applications to small area estimation. Test, 20, 574-588.

Deville, J.-C., and Särndal, C.-E. (1992). Calibration estimators in survey sampling. Journal of the American Statistical Association, 87, 376-382.

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

Hidiroglou, M.A., and Estevao, V.M. (2016). A comparison of small area and traditional estimators via simulation. Statistics in Transition, 17, 133-154.

Huang, R., and Hidiroglou, M.A. (2003). Design consistent estimators for a mixed linear model on survey data. Proceedings of the Section on Survey Research Methods, American Statistical Association, 1897-1904.

Moghtased-Azar, K., Tehranchi, R. and Amiri-Simkooei, A.R. (2014). An alternative method for non-negative estimation of variance components. Journal of Geodesy, 88, 427-439.

Nandram, B., and Sayit, H. (2011). A Bayesian analysis of small area probabilities under a constraint. Survey Methodology, 37, 2, 137-152. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2011002/article/11603-eng.pdf.

Pfeffermann, D., and Barnard, C. (1991). Some new estimators for small area means with applications to the assessment of farmland values. Journal of Business and Economic Statistics, 9, 73-83.

Prasad, N.G.N., and Rao, J.N.K. (1990). The estimation of the mean squared error of small-area estimators. Journal of the American Statistical Association, 85, 163-171.

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

Särndal, C.-E., Swensson, B. and Wretman, J.H. (1989). The weighted residual technique for estimating the variance of the general regression estimator of the finite population total. Biometrika, 76, 527-537.

Stefan, M., and Hidiroglou, M.A. (2020). Benchmarked estimators for a small area mean under a one-fold nested regression model. International Statistical Review, (To appear).

Tillé, Y. (2006). Sampling Algorithms. New York: Springer.

Ugarte, M.D., Militino, A.F. and Goicoa, T. (2009). Benchmarked estimates in small areas using linear mixed models with restrictions. Test, 18, 342-364.

Verret, F., Rao, J.N.K. and Hidiroglou, M.A. (2015). Model-based small area estimation under informative sampling. Survey Methodology, 41, 2, 333-347. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2015002/article/14248-eng.pdf.

Wang, J., Fuller, W.A. and Qu, Y. (2008). Small area estimation under a restriction. Survey Methodology, 34, 1, 29-36. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2008001/article/10619-eng.pdf.

You, Y., and Rao, J.N.K. (2002). A pseudo-empirical best linear unbiased prediction approach to small area estimation using survey weights. The Canadian Journal of Statistics, 30, 431-439.

You, Y., Rao, J.N.K. and Dick, P. (2004). Benchmarking hierarchical Bayes small area estimators in the Canadian census undercoverage estimation. Statistics in Transition, 6, 631-640.

You, Y., Rao, J.N.K. and Hidiroglou, M.A. (2013). On the performance of self benchmarked small area estimators under the Fay-Herriot area level model. Survey Methodology, 39, 1, 217-229. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2013001/article/11830-eng.pdf.

• Previous
• Article main page
• Next