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

研究提出了各种问题的统计 决策理论 本研究的主要重点是 以容许性为一的推理方法的评价 评价标准 可受理性与 形式贝叶斯规则和对称马尔可夫递归 链形成了理论背景的重要部分 为了这项工作。 在非参数设置中， 的诱导推理。 这种选择很容易 数字实现，并与两个频率论 和贝叶斯自举过程。 另一个建议领域 研究涉及的问题，渐近 特征值、特征投影和奇异分布 价值观 以前的工作包含新的技术， 特征值和奇异值的统一处理。 这些结果的具体例子和推广应用 给出了特征投影问题的基本结果。 统计学研究的是发表声明的活动 根据现有的样本或数据对人口进行分析。 由于信息往往是不完整的，这些声明是 通常是错误的。 这个错误的量化是一个 标准的统计活动，通常用 概率 该基金支持的研究涉及 各种量化统计误差的方法， 这些方法的特性。 研究的最终目的是 开发新的统计方法， 推理陈述中错误的概率量化。

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.