Survey Methodology
Local polynomial estimation for a small area mean under informative sampling

by Marius Stefan and Michael A. Hidiroglou^{Note 1}

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Abstract

Model-based methods are required to estimate small area parameters of interest, such as totals and means, when traditional direct estimation methods cannot provide adequate precision. Unit level and area level models are the most commonly used ones in practice. In the case of the unit level model, efficient model-based estimators can be obtained if the sample design is such that the sample and population models coincide: that is, the sampling design is non-informative for the model. If on the other hand, the sampling design is informative for the model, the selection probabilities will be related to the variable of interest, even after conditioning on the available auxiliary data. This will imply that the population model no longer holds for the sample. Pfeffermann and Sverchkov (2007) used the relationships between the population and sample distribution of the study variable to obtain approximately unbiased semi-parametric predictors of the area means under informative sampling schemes. Their procedure is valid for both sampled and non-sampled areas. Verret, Rao and Hidiroglou (2015) studied alternative procedures that incorporate a suitable function of the unit selection probabilities as an additional auxiliary variable. Their procedure resulted in approximately unbiased empirical best linear unbiased prediction (EBLUP) estimators for the small area means. In this paper, we extend the Verret et al. (2015) procedure by not assuming anything about the inclusion probabilities. Rather, we incorporate them into the unit level model via a smooth function of the inclusion probabilities. This function is estimated via a local approximation resulting in a local polynomial estimator. A conditional bootstrap method is proposed for the estimation of mean squared error (MSE) of the local polynomial and EBLUP estimators. The bias and efficiency properties of the local polynomial estimator are investigated via a simulation. Results for the bootstrap estimator of MSE are also presented.

Key Words:      Local polynomial estimation; EBLUP estimation; Augmented model; Nested error model; Informative sampling; Conditional bootstrap.

Table of contents

• Section 1. Introduction
• Section 2. Existing methods
• Section 3. The local polynomial estimator
• Section 4. MSE estimation based on the bootstrap
• Section 5. Simulation study
• Section 6. Concluding remarks
• Acknowledgements
• References

How to cite

Stefan, M., and Hidiroglou, M.A. (2020). Local polynomial estimation for a small area mean under informative sampling. Survey Methodology, Statistics Canada, Catalogue No. 12-001-X, Vol. 46, No. 1. Paper available at http://www.statcan.gc.ca/pub/12-001-x/2020001/article/00002-eng.htm.

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