A method to find an efficient and robust sampling strategy under model uncertainty
Section 7. Conclusions

• Previous
• Next

The strategy that couples $\pi \text{ps}$ with the difference estimator is optimal when the parameters of the superpopulation model are known. Taking into account that these assumptions are seldom satisfied, it was shown in Section 3 and illustrated in Subsection 6.1 that this optimality breaks down even under small misspecifications of the model.

In Section 4 we propose a method for choosing the sampling design, which is extended to its use with the GREG estimator in Section 5. The method allows for taking the uncertainty about the model parameters into account by introducing a prior distribution on them. Although it could be argued that a source of subjectivity is added by introducing a prior distribution on the parameters, our view is that it is more subjective to choose the design without any type of assessment of the assumptions. Furthermore, inference is still design-based, as the prior is used only for choosing the design.

The method was illustrated with a real dataset, yielding satisfactory results. It should be noted that although the illustrations used stratified simple random sampling, the method in this article is valid for any sampling design.

Appendix

Proof of (4.2)

Proof. The following expectations are required in the proof,

$${\text{E}}_{\xi }{Y}_k\mathrm{<mo>=</mo>}{\text{E}}_{\xi }\left[f\left(x_k|{\beta }_1\right)+{\epsilon }_k\right]\mathrm{<mo>=</mo>}f\left(x_k|{\beta }_1\right)(\text{A}\text{.1})$$

$${\text{E}}_{\xi }{Y}_k^2\mathrm{<mo>=</mo>}{\text{E}}_{\xi }\left[{\left(f\left(x_k|{\beta }_1\right)+{\epsilon }_k\right)}^2\right]\mathrm{<mo>=</mo>}f{\left(x_k|{\beta }_1\right)}^2+{\sigma }^{2}g{\left(x_k|{\beta }_2\right)}^2(\text{A}.2)$$

${\text{E}}_{\xi }\overline{Y},$ ${\text{E}}_{\xi }{\overline{Y}}^2$ and ${\text{E}}_{\xi }\overline{f\text{}Y}$ are obtained using (A.1) and (A.2),

$${\text{E}}_{\xi }\overline{Y}\mathrm{<mo>=</mo>}{\text{E}}_{\xi }\left[\frac{1}{N}{\displaystyle \sum _U{Y}_k}\right]\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _U{\text{E}}_{\xi }{Y}_k}\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{U}f\left(x_k|{\beta }_1\right)}\equiv \overline{f}(\text{A}.3)$$

$${\text{E}}_{\xi }{\overline{Y}}^2\mathrm{<mo>=</mo>}{\text{E}}_{\xi }\left[\frac{1}{N}{\displaystyle \sum _U{Y}_k^2}\right]\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _U\left(f{\left(x_k|{\beta }_1\right)}^2+{\sigma }^{2}g{\left(x_k|{\beta }_2\right)}^2\right)}\equiv {\overline{f}}^2+{\sigma }^2{\overline{g}}^2(\text{A}.4)$$

$${\text{E}}_{\xi }\overline{f\text{}Y}\mathrm{<mo>=</mo>}{\text{E}}_{\xi }\left[\frac{1}{N}{\displaystyle \sum _{U}f\left(x_k|\beta \right)Y_k}\right]\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{U}f\left(x_k|\beta \right){\text{E}}_{\xi }{Y}_k}\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{U}f{\left(x_k|\beta \right)}^2}\mathrm{<mo>=</mo>}{\overline{f}}^2.(\text{A}.5)$$

Now, using (A.3), (A.4) and (A.5) we get

$${\text{E}}_{\xi }\left[\overline{f\text{}Y}-\overline{f\text{}Y}\right]\mathrm{<mo>=</mo>}{\overline{f}}^2-{\overline{f}}^2\mathrm{<mo>=</mo>}{S}_{f\mathrm{,}f}(\text{A}.6)$$

$${\text{E}}_{\xi }\left[{\overline{Y}}^2-{\overline{Y}}^2\right]\mathrm{<mo>=</mo>}{\overline{f}}^2+{\sigma }^2{\overline{g}}^2-{\overline{f}}^2\mathrm{<mo>=</mo>}{S}_{f\mathrm{,}f}+{\sigma }^2{\overline{g}}^2.(\text{A}.7)$$

Using (A.6) and (A.7), we obtain an approximation to the correlation coefficient, $R_{f\mathrm{,}y},$

$$R_{f\mathrm{,}y}^2\mathrm{<mo>=</mo>}\frac{{\left(\overline{f\text{}y}-\overline{f\text{}y}\right)}^2}{\left({\overline{f}}^2-{\overline{f}}^2\right)\left({\overline{y}}^2-{\overline{y}}^2\right)}\approx \frac{{\text{E}}_{\xi }^2\left[\overline{f\text{}Y}-\overline{f\text{}Y}\right]}{{\text{E}}_{\xi }\left[\left({\overline{f}}^2-{\overline{f}}^2\right)\left({\overline{Y}}^2-{\overline{Y}}^2\right)\right]}\mathrm{<mo>=</mo>}\frac{S_{f\mathrm{,}f}}{S_{f\mathrm{,}f}+{\sigma }^2{\overline{g}}^2}.(\text{A}.8)$$

Solving (A.8) for ${\sigma }^2$ we get (4.2), as desired. The proof of (5.6) is analogous.

References

Beaumont, J.-F., Haziza, D. and Ruiz-Gazen, A. (2013). A unified approach to robust estimation in finite population sampling. Biometrika, 100, 3, 555-569.

Bramati, M. (2012). Robust Lavallée-Hidiroglou stratified sampling strategy. Survey Research Methods, 6, 3, 137-143.

Cassel, C.M., Särndal, C.-E. and Wretman, J. (1976). Some results on generalized difference estimation and generalized regression estimation for finite populations. Biometrika, 63, 3, 615-620.

Cassel, C.M., Särndal, C.-E. and Wretman, J. (1977). Foundations of Inference in Survey Sampling. New York: John Wiley & Sons, Inc.

Dalenius, T., and Hodges, J.L. (1959). Minimum variance stratification. Journal of the American Statistical Association, 54, 88-101.

Godambe, V.P. (1955). A unified theory of sampling from finite populations. Journal of the Royal Statistical Society, Series B, 17, 269-278.

Hájek, J. (1959). Optimal strategy and other problems in probability sampling. Casopis Pro Pestování Matematiky, 84, 4, 387-423.

Holmberg, A., and Swensson, B. (2001). On pareto $\pi \text{ps}$ sampling: Reflections on unequal probability sampling strategies. Theory of Stochastic Processes, 7(23), 142-155.

Horvitz, D.G., and Thompson, D.J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 260, 663-685.

Isaki, C.T., and Fuller, W.A. (1982). Survey design under the regression superpopulation model. Journal of the American Statistical Association, 77, 89-96.

Lanke, J. (1973). On UMV-estimators in survey sampling. Metrika, 20, 196-202.

Narasimhan, B., Johnson, S., Hahn, T., Bouvier, A. and Kiêu, K. (2019). Cubature: Adaptive Multivariate Integration Over Hypercubes. R package version 2.0.4. https://CRAN.R-project.org/package=cubature.

Nedyalkova, D., and Tillé, Y. (2008). Optimal sampling and estimation strategies under the linear model. Biometrika, 95, 3, 521-537.

R Core Team (2020). R: A language and environment for statistical computing. The R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.

Rosén, B. (1997). On sampling with probability proportional to size. Journal of Statistical Planning and Inference, 62, 159-191.

Rosén, B. (2000). Generalized Regression Estimation and Pareto $\pi \text{ps}\text{.}$ R&D Report 2000:5. Statistics Sweden.

Royall, R.M., and Herson, J. (1973). Robust estimation in finite populations I. Journal of the American Statistical Association, 68, 344, 880-889.

Särndal, C.-E., Swensson, B. and Wretman, J. (1992). Model Assisted Survey Sampling. Springer.

Tillé, Y., and Wilhelm, M. (2017). Probability sampling designs: Principles for choice of design and balancing. Statistical Science, 32(2), 176-189.

Wright, R.L. (1983). Finite population sampling with multivariate auxiliary information. Journal of the American Statistical Association, 78, 879-884.

Zhai, Z., and Wiens, D. (2015). Robust model-based stratification sampling designs. The Canadian Journal of Statistics, 43, 4, 554-577.

• Previous
• Article main page
• Next