Local polynomial estimation for a small area mean under informative sampling
Section 6. Concluding remarks

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In this paper, we studied the estimation of a small area mean under informative sampling by using an augmented model approach where the augmenting variable is a smooth function $m_0\left(p_{j|i}\right)$ of the selection probability $p_{j|i}.$ Our augmented model is semi-parametric. It differs from Verret et al. (2015), in that nothing was assumed about the augmenting function $m_0(\cdot ).$

We proposed a three-step procedure to estimate the augmented semi-parametric model. Firstly, local polynomial fits were estimated for each unit of the population (sampled and non-sampled). Secondly, given these local fits a new dependent variable was defined to obtain global estimators of the regression parameters and the small area effects. The resulting estimators were used to compute the predicted values of the dependent variable, $y,$ for all non-sampled units. Finally, using the observed sample values of $y,$ and the predicted values of $y,$ we computed the local polynomial estimator ${\widehat{\overline{Y}}}_i^{\text{LP}}$ for the small area mean  ${\overline{Y}}_i.$

We adopted the conditional parametric bootstrap method to estimate the mean squared error of the newly proposed estimator. The conditional bootstrap is a modified version of the parametric bootstrap estimator method of Hall and Maiti (2006).

We carried out a simulation study to compare the bias and mean squared error performance of the usual EBLUP, ${\widehat{\overline{Y}}}_i^{\text{EBLUP}},$ the augmented EBLUP of Verret et al. (2015), ${\widehat{\overline{Y}}}_i^{\text{VRH}},$ and the proposed local polynomial estimator, ${\widehat{\overline{Y}}}_i^{\text{LP}}.$ As expected, ${\widehat{\overline{Y}}}_i^{\text{EBLUP}}$ exhibited large bias under informative sampling. The new estimator ${\widehat{\overline{Y}}}_i^{\text{LP}}$ had equal or smaller MSE than ${\widehat{\overline{Y}}}_i^{\text{VRH}}$ when the sample design was highly informative. If the sample design is less informative, it is better to use one of the two estimators in Verret et al. (2015): that is, augment the basic model with either $p_{j|i}$ or $\log \left(p_{j|i}\right).$ Note that in doing so, the gains are very small. If the sampling design is very slightly or not at all informative, then estimator ${\widehat{\overline{Y}}}_i^{\text{EBLUP}}$ based on the population model should be used.

We also evaluated the performance of the mean squared error bootstrap estimation for the estimators ${\widehat{\overline{Y}}}_i^{\text{EBLUP}},$ ${\widehat{\overline{Y}}}_i^{\text{VRH}}$ and ${\widehat{\overline{Y}}}_i^{\text{LP}},$ in terms of average absolute relative bias $\left(\overline{\text{ARB}}\right)$ and average confidence level $\left(\overline{\text{CL}}\right).$ The conditional bootstrap provides a good way to estimate the mean squared errors.

The advantage of the local polynomial approach is that it provides an automatic way of augmenting the model when the design is informative. Its biggest disadvantage is its computational burden both in terms of parameter estimation and associated reliability. The procedure outlined in Section 5.3 suggests a way to determine whether it is worth using it or not. An alternative approach is to augment the unit level model with a P-spline term of selection probabilities to account for the informativeness of the sampling design. This approach has been recently studied by Chatrchi (2018).

Acknowledgements

We would like to thank J.N.K. Rao for suggesting the conditional bootstrap for estimating the mean squared error of the polynomial estimator, and commenting on the article. We would also like to thank the Associate Editor and a referee for their constructive comments that have improved the paper.

References

Asparouhov, T. (2006). General multi-level modelling with sampling weights. Communication in Statistics, Theory and Methods, 439-460.

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.

Breidt, F.J., and Opsomer, J.D. (2000). Local polynomial regression estimators in survey sampling. Annals of Statistics, 28, 4, 1026-1053.

Breidt, F.J., Opsomer, J.D., Johnson, A.A. and Ranalli, M.G. (2007). Semiparametric model-assisted estimation for natural resource surveys. Survey Methodology, 33, 1, 35-44. Paper available at https://www150.statcan.gc.ca/n1/pub/12-001-x/2007001/article/9850-eng.pdf.

Chatrchi, G. (2018). Small Area Estimation: Informative Sampling and Two-Fold Models. Unpublished Ph.D. Thesis, Carleton University , Ottawa , Canada .

Fan, J., and Gijbels, I. (1996). Local Polynomial Modelling and its Applications. London: Chapman and Hall.

González-Manteiga, W., Lombardia, M.J., Molina, I. , Morales, D. and Santamaria, L. (2008). Bootstrap mean squared error of a small area EBLUP. Journal of Statistical Computation and Simulation, 78, 5, 443-462.

Hall, P., and Maiti, T. (2006). On parametric bootstrap methods for small area prediction. Journal of the Royal Statistical Society, Series B: Statistical Methodology, 68, 221-238.

Opsomer, J.D., Claeskens, G., Ranalli, M.G., Kauermann, G. and Breidt, F.J. (2008). Nonparametric small area estimation using penalized spline regression. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 70(1), 265-286.

Pfeffermann, D., and Sverchkov, M. (2007). Small-area estimation under informative probability sampling of areas and within the selected areas. Journal of the American Statistical Association, 102, 480, 1427-1439.

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, 409, 163-171.

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

Rao, J.N.K., Sinha, S.K. and Dumitrescu, L. (2014). Robust small area estimation under semi-parametric mixed models. The Canadian Journal of Statistics, 42, 126-141.

Ruppert, D., and Matteson, D.E. (2015). Statistics and Data Analysis for Financial Engineering with R Examples: 2^nd Ed., New York : Springer.

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

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/pub/12-001-x/2015002/article/14248-eng.pdf.

Wu, H., and Zhang, J.T. (2002). Local polynomial mixed-effects models for longitudinal data. Journal of the American Statistical Association, 97, 459, 883-897.

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