Parametric and Semiparametric Bayesian Methods for Small Area Estimation

本文主要研究小区域估计的参数和半参数递阶贝叶斯方法。对当地地区(如县、市或人口普查分区)的直接调查估计通常伴随着较大的标准误差和变异系数，因为这些地区的样本量很小。这样做的主要原因是，最初的调查旨在实现比当地地区高得多的总体水平的准确性。这就使得有必要从类似的地方“借力”或利用信息。分层贝叶斯方法和经验贝叶斯方法特别适合于满足这一需求。然而，许多贝叶斯文献仅限于正态理论小区域均值的分层和经验贝叶斯估计以及其他感兴趣的特征，并且假设局部区域效应的正态分布，这也几乎完全是参数的。这项研究的主要组成部分之一是发展半参数分层贝叶斯方法，避免假设局部区域效应的正态分布。这些方法既适用于线性模型，也适用于广义线性模型。这些方法的具体应用将包括估计四人家庭的收入中位数，估计县、县以下、人口普查区域等小地方的收入和贫困情况。该方法也适用于其他问题，既可以用于离散数据的分析，也可以用于连续数据的分析。这项研究的第二个方面是对更复杂的调查进行小区域估计，例如分层两阶段抽样，其中局部区域由跨越地层边界的基本单元组成。这里将采用参数和半参数分层贝叶斯方法，并对结果进行比较。

This research focuses on the development of parametric and semiparametric hierarchical Bayesian methods for small area estimation. Direct survey estimates of local areas (such as a county, municipality, or census division) are usually accompanied by large standard errors and coefficients of variation due to smallness of samples sizes in these areas. The prime reason for this is that the original survey was targeted to achieve accuracy at a much higher level of aggregation than that for local areas. This makes it a necessity to `borrow strength` or use information from similar local areas. Hierarchical and empirical Bayes methods are particularly well-suited to meet this need. Much of the Bayesian literature, however, is restricted to normal theory hierarchical and empirical Bayes estimation of small area means and other characteristics of interest, and that too has almost exclusively been parametric, assuming normality of the local area effects. One of the major components of this research is the development of semiparametric hierarchical Bayesian methods that avoid assuming normality of the local area effects. These methods are applicable to both linear and generalized linear models. Specific applications of these methods will involve estimation of median income of four-person families, estimation of income and poverty for small places like counties, subcounties, census tracts, etc. The methodology is applicable to other problems and can be used for the analysis of both discrete and continuous data. The second aspect of this research is small area estimation for more complex surveys, such as stratified two-stage sampling, where the local areas consist of primary units which cut across the stratum boundaries. Both parametric and semiparametric hierarchical Bayesian methods will be pursued here and the results will be compared.