Mathematical Sciences: Inference Problems in Multivariate Statistical Analysis

本研究关注的是推理问题， 经典的多变量程序，例如多变量分析， 方差分析，判别分析，主成分 分析和典型相关分析。 的问题 构成有两种主要类型。 一个是估算的问题 典型相关和参数的特征结构 在多变量方差分析中出现的矩阵。 的 研究人员建议应用决策理论概念， 评估基于这些估计的属性和行为 理念的 第二类要调查的问题涉及 假设检验 多变量分析中广泛使用的检验 是那些假设平等的一个子集， 参数矩阵的特征根。 没有小样本 性质（无偏性，幂函数的单调性， （可受理性等）都以常见的测试而闻名，包括 似然比检验 自助的用处 多元分析中推理问题中技术将 追究 经典的多变量统计程序是 以线性、独立和 常态 这些程序在整个 科学分析实验和观测数据。 的 统计程序，以作出一些估计所使用的 这些多变量分析是粗略的，因此， 它们可能会对科学家的结论产生重大影响， 画出测量之间的关系。 也是在这一 设置，我们目前对性能的知识 统计决策程序严格适用于 重复了很多次的实验。 由于重复实验通常是昂贵的， 通常以低频完成。 研究发现， 统计决策程序的执行情况， 重复次数较少，预计可提供有价值的 信息和洞察力，以广泛的科学 社区

This research is concerned with problems of inference in classical multivariate procedures such as multivariate analysis of variance, discriminant analysis , principal components analysis, and canonical correlations analysis. The problems posed are of two main types. One involves problems in estimating canonical correlations and the eigenstructures of parameter matrices arising in multivariate analysis of variance. The investigators propose to apply decision-theoretic concepts and to evaluate the properties and behavior of estimates based on these concepts. The second type of problem to be investigated involves hypothesis testing. Widely used tests in multivariate analysis are ones which hypothesize the equality of a subset of the characteristic roots of a parameter matrix. No small sample properties (unbiasedness, monotonicity of the power functions, admissibility, etc.) are known for common tests, including the likelihood ratio test. The usefulness of the bootstrapping technique in problems of inference in multivariate analysis will be investigated. Classical multivariate statistical procedures are characterized by assumptions of linearity, independence and normality. These procedures are widely used throughout the sciences to analyze experimental and observational data. The statistical procedures for making some of the estimates used in these multivariate analyses are crude and, therefore, improving them may have a significant impact on the conclusions scientists draw about relationships between measurements. Also in this setting, our current knowledge about the performance of statistical decision making procedures applies strictly to experiments that are repeated a very large number of times. Since replicating experiments is usually expensive, it is typically done with low frequency. The research into the performance of statistical decision making procedures when the number of replicates is small is expected to provide valuable information and insight to a broad spectrum of the scientific community.