子总体最优等精度样本分配的特征问题方法
An eigenproblem approach to optimal equal-precision sample allocation in subpopulations
摘要：
分层和/或多级抽样中的样本分配问题是抽样论的核心问题之一。在总体调查中，子总体参数估计量的精度约束往往必须在样本分配过程中加以考虑。这些问题通常用数学规划程序来解决。在许多情况下，最好是分配样本，以迫使在亚总体一级的估计精度是最佳的和相同的，而样本（或样本，在两阶段抽样）的总（预期）大小的限制是强加的。这里我们主要关注的是两阶段抽样方案。我们表明，这样的问题在广泛的一类抽样计划有一个优雅的数学和计算的解决方案。这是由于最优化问题的适当定义，这使得能够通过线性代数设置来解决它，该设置涉及根据某些种群数量定义的矩阵的特征值和特征向量。作为最终结果，我们得到了一个非常简单和相对通用的方法来计算子种群的最优和等精度分配，这是基于线性代数的最标准的算法之一（例如，在R软件中可用）。理论上的解决方案说明通过一个数值例子的基础上劳动力调查。最后，我们要强调的是，我们所描述的方法，允许相当自动地适应不同级别的精度优先级的亚群。
英文摘要：
Allocation of samples in stratified and/or multistage sampling is one of the central issues of sampling theory. In a survey of a population often the constraints for precision of estimators of subpopulations parameters have to be taken care of during the allocation of the sample. Such issues are often solved with mathematical programming procedures. In many situations it is desirable to allocate the sample, in a way which forces the precision of estimates at the subpopulations level to be both: optimal and identical, while the constraints of the total (expected) size of the sample (or samples, in two-stage sampling) are imposed. Here our main concern is related to two-stage sampling schemes. We show that such problem in a wide class of sampling plans has an elegant mathematical and computational solution. This is done due to a suitable definition of the optimization problem, which enables to solve it through a linear algebra setting involving eigenvalues and eigenvectors of matrices defined in terms of some population quantities. As a final result we obtain a very simple and relatively universal method for calculating the subpopulation optimal and equal-precision allocation which is based on one of the most standard algorithms of linear algebra (available e.g. in R software). Theoretical solutions are illustrated through a numerical example based on the Labour Force Survey. Finally, we would like to stress that the method we describe, allows to accommodate quite automatically for different levels of precision priority for subpopulations.
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