Sample allocation for efficient model-based small area estimation
Section 1. Introduction
In this paper we present a new model-based allocation method in stratified sampling where the areas of interest coincide with the strata. Our study is focused on the components of an efficient area allocation. A clear starting point for the allocation process is reached if the areas of interest are defined as early as in the design phase of the research and if it is also known how large a sample is allowed in consideration of the disposable resources (time, budget etc.). The choice of the allocation method depends on various factors such as the selected model, estimation method, available pre-information of the population and the optimization criteria set only on area or population level, or on both levels simultaneously.
We have selected six existing allocation methods and developed a new one which we call a model-based allocation. The general properties of these methods are examined in Section 2 and Section 3. Five of these allocations can be regarded as model-free. Two of them use only number-based information, such as the number of areas and the number of basic units in each area. Three other allocations need, in addition to number-based information, area level parameter information, such as area totals, standard deviation or coefficient of variation (CV). Because this information about the study variable is not available, a common solution is to replace it with a proper proxy variable. The last of the reference allocations, introduced by Molefe and Clark (MC) (2015), is a model-assisted allocation which is based on a composite estimator and a two-level model. We have named it MC-allocation.
The optimization criteria of the five model-free allocations differ from one another. Allocations based only on area-specific numbers can be computed easily, but their choice is reasonable under limited circumstances. In each of the parameter-based allocations the optimization criterion is different. It can be set on the level of the population parameter estimate (Neyman allocation) or on area level estimates in average (Bankier allocation). The third allocation solution, which deviates from the two former ones, is the NLP allocation, in which the tolerances of estimates are set on both population and area level.
This article starts from the assumption that if model-assisted or model-based estimation is used in a survey the model and estimation method must be taken into account when the allocation of the sample into areas is designed. This was used as a starting point when the new model-based allocation, presented in Section 2, was derived. Also, one of the reference allocations, model-assisted allocation, is based on a given model.
The comparison of performances of different allocation methods in real situations has been implemented by using simulation experiments and is presented in Section 4. An official Finnish register of block apartments for sale serves as the population. The structure of the register is introduced in Section 4.1. An auxiliary variable has been used in place of the study variable when computing the area sample sizes for each allocation except equal and proportional allocation. The comparison demonstrates clearly that these allocations lead to different sample distributions. The same kind of variety also concerns their performances. We have applied model-based EBLUP (Empirical Best Linear Unbiased Predictor) estimation on the allocations when estimating the area totals of the study variable. For measuring and comparing the performances of allocations, a relative root mean square error RRMSE% and absolute relative bias ARB% were used.
In Section 5 empirical simulation results are discussed as concluding remarks. They support the allocation solution in which not only auxiliary information, but also the model and estimation method should be determined as early as in the design phase of a survey. A good example is the allocation presented in Section 2.2. The most accurate area estimates of area totals were obtained by using this method.
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