Mathematical Sciences: Some Bayesian Problems in Sample Survey

这项建议将考虑将贝叶斯方法应用于有限总体抽样中的一些问题。在过去的几年里，首席调查员帮助开发了一种非信息性的贝叶斯方法来解决有限总体抽样中的一些问题。它是建立在“Polya后验”的基础上的，当一个人对单位的先验信念大致可交换时，它是适当的。该项目的第一个主要目标是将作为“波利亚后验”基础的技术推广到整群抽样领域。这有望为一类重要的问题提供一个灵活而统一的理论。马尔可夫链蒙特卡罗技术的最新发展意味着，直到最近还在计算上难以处理的贝叶斯模型现在可以研究了。这个项目的第二个主要目标是使用这些方法来开发贝叶斯模型，当一个人对单位的先前信念不再可以交换时，用于调查抽样。在要考虑的一类模型中，单元将被假定为部分可交换的，而在另一类中，总体将被假定由广义的骨灰盒过程产生。这应该产生一个模型集合，扩展可用于抽样调查分析的先验信息的类型。调查抽样或有限总体抽样中的标准设定假设某一总体的每个成员都具有未知数量的感兴趣的特征。统计学家感兴趣的是估计与这一感兴趣的特征有关的数量。例如，利息的特征可以是家庭收入，给定城市的所有家庭的人口，以及该城市的平均或中位数家庭收入的利息量。然后，统计学家将使用从人口中抽取的样本中的特征值来估计这一数量。这个项目的主要目标是研究如何利用关于人口的先验信息来构建对感兴趣的数量的合理估计。它将利用计算机模拟的最新发展来研究各种模型，以获取到目前为止无法研究的先验信息。

This proposal will consider the application of the Bayesian methodology to some problems in finite population sampling. Over the past several years the principle investigator has helped to develop a noninformative Bayesian approach to some problems in finite population sampling. It is based on the `Polya posterior' and is appropriate when one's prior beliefs about the units are roughly exchangeable. The first major goal of this project is to extend the techniques underlying the `Polya posterior' to the area of cluster sampling. This promises to yield a flexible and unified theory for an important class of problems. Recent developments in Markov chain Monte Carlo technology mean that Bayesian models which until recently have been computationally intractable can now be studied. The second major goal of this project is to use these methods to develop Bayesian models for survey sampling when one's prior beliefs about the units are no longer exchangeable. In one class of models to be considered the units will be assumed to be partially exchangeable, while in another class the populations will be assumed to be generated by a generalized urn processes. This should yield a collection of models that extends the type of prior information that can be used in an analysis in sample survey. A standard setting in survey sampling or finite population sampling assumes each member of some population possesses an unknown amount of a characteristic of interest. The statistician is interested in estimating a quantity related to this characteristic of interest. For example the characteristic of interest could be household income, the population all households in a given city and the quantity of interest the average or median household income for the city. The statistician will then use the values of the characteristic in a sample drawn from the population to estimate this quantity. The major goal of this project is to study how prior information about the population can be used to construct sensible estimates about the quantity of interest. It will use recent developments in computer simulation to investigate various models for prior information which up until now have been impossible to study.