Mathematical Sciences: Statistical Inferences, Decision Theory, and Asymptotics of Eigenvalues and Eigenprojections

对统计决策理论中的各种问题进行了研究。本研究的重点是以可采性为评价标准的推理方法的评价。形式贝叶斯规则的可容许性与对称马氏链的常返性之间的联系构成了这项工作的重要理论背景。在非参数环境下，提出了一种可供选择的推理方法。这种选择在数值上很容易实现，并且与频域和贝叶斯自举过程都有联系。另一个拟议的研究领域涉及与特征值、特征投影和奇异值的渐近分布有关的问题。以前的工作包含了新的技术，它们提供了特征值和奇异值的统一处理。给出了这些结果在具体例子中的应用，并将基本结果推广到特征投影问题。统计学研究的是根据现有的样本或数据对人口进行陈述的活动。由于信息往往是不完整的，这种说法通常是错误的。这种误差的量化是一种标准的统计活动，通常用概率来表示。这笔赠款支持的研究涉及量化统计误差的各种方法以及这些方法的性质。这项研究的最终目标是开发新的统计方法，以获得推理语句中错误的概率量化。

Research is proposed on a variety of problems in statistical decision theory. The main focus of this research is the evaluation of inferential methods with admissibility as one evaluative criterion. The connection between the admissibility of formal Bayes rules and the recurrence of symmetric Markov chains forms an important portion of the theoretical background for this work. In a nonparametric setting an alternative method of inducing inferences is proposed. This alternative is easily implemented numerically and has connections with both frequentist and Bayesian bootstrap procedures. Another area of proposed research concerns questions related to the asymptotic distribution of eigenvalues, eigenprojections, and singular values. Previous work contains new techniques which provide a unified treatment of eigenalues and singular values. Applications of these results to concrete examples and extension of the basic results to eigenprojection problems is proposed. Statistics deals with the activity of making statements about populations on the basis of available samples or data. Since information is most often incomplete, such statements are typically in error. The quantification of this error is a standard statistical activity and is often expressed in terms of probability. The research supported by this grant concerns various methods for quantifying statistical errors and the properties of such methods. The ultimate goal of the research is the development of new statistical methods for obtaining probabilistic quantification of errors in inferential statements.