A Noninformative Bayesian Approach to some Finite Population Problems when Auxiliary Variables are Present

9971331在有限总体抽样中，除了感兴趣的特征之外，通常还存在包含有关特征的信息的辅助变量。 例如，可以先验地知道这些辅助变量的总体均值或某些总体分位数。 经典理论已经开发出各种各样的方法来利用这些信息，具有不同程度的整体有效性。 波利亚后验是作为一种无信息贝叶斯方法，当很少或没有先验信息时，有限的人口抽样。 所提出的研究的目标是扩大波利亚后辅助变量存在的问题。 这将通过以自然的方式限制Polya后验来完成，使得给定样本，可以生成整个群体的模拟副本，其满足由关于辅助变量的先验知识引起的约束。 然后，可以使用通常的贝叶斯方式使用模拟群体进行统计推断。两个主要的技术问题将是发展基本的理论和技术，用于实施与现有信息一致的模拟。这就允许对标准理论现在必须个别考虑的问题采取一种连贯的方法。 几个问题将被确定为某些类型的先验信息，目前往往被忽视，可以在一个客观和有效的方式使用。 统计学的一个基本问题是，当信息只存在于构成总体的一个样本或子集中时，如何对总体进行推断。 例如，我们可能想估计在给定的行业中，一个典型的工人在一年中因病缺勤的天数。 除了样本的资料外，我们亦可知道该行业所有工人的平均年龄或教育水平中位数。 统计学家已经开发了各种方法来处理这些问题，这取决于手头有什么样的样本之外的信息。 有些方法比其他方法效果好得多。在本研究中，我们将研究一种通用的方法，可以有效地利用各种类型的信息。这是通过使用样本中的数据和先验信息来构建与样本数据和先验信息都一致的整个总体的模拟或随机副本来完成的。 通过考虑这些随机生成的总体副本之间的变异性，人们不仅可以找到对感兴趣的数量的估计，而且可以找到与估计相关的不确定性的度量。

9971331In finite population sampling in addition to the characteristic of interest, there are often auxiliary variables present which contain information about the characteristic. For example, a priori the population means or some of the population quantiles for these auxiliary variables could be known. Classical theory has developed a variety of methods to exploit this information with different levels of overall effectiveness. The Polya posterior was developed as a noninformative Bayesian approach to finite population sampling when little or no prior information is available. The goal of the proposed research is to extend the Polya posterior to problems where auxiliary variables are present. This will be done by restricting the Polya posterior in a natural way so that given a sample simulated copies of the entire population can be generated which satisfy the constraints induced by the prior knowledge about the auxiliary variables. Statistical inference can then be carried out using the simulated populations in the usual Bayesian manner. The two main technical problems will be to develop the underlying theory and the techniques for implementing the simulations which are consistent with the prior information at hand. This allows for a coherent approach to problems which standard theory must now consider individually. Several problems will be identified were certain types of prior information which are often presently ignored can be used in an objective and effective way. The resulting noninformative Bayesian procedures should have good frequentist properties.A fundamental problem of statistics is making inferences about a population when information is available only about a sample or subset of the individuals making up the population. For example, we may want to estimate how many days a typical worker is absent because of illness during a year in a given industry. In addition to the information in the sample, we may also know the average age or median education level for all the workers in the industry. Statisticians have developed various methods to handle such problems depending on what kind of information beyond the sample is at hand. Some methods work much better than others. In this research we will study a general approach which can effectively make use of a variety of types of information. This is done by using the data in the sample and the prior information to construct simulated or random copies of the entire population which are consistent with both the sample data and the prior information. By considering the variability among these randomly generated copies of the population one can find not only an estimate of the quantity of interest but a measure of uncertainty associated with the estimate.