Mathematical Sciences: Inference Problems in Multivariate Statistical Analysis

本研究主要针对经典多变量分析中的推理问题，如多变量方差分析、判别分析、主成分分析和典型相关分析等。提出的问题主要有两类。其中一个涉及在多元方差分析中出现的估计典型相关和参数矩阵的特征结构的问题。研究者建议运用决策理论的概念，并根据这些概念来评估评估的性质和行为。要研究的第二类问题涉及假设检验。在多变量分析中广泛使用的检验是假设参数矩阵的特征根的子集相等的检验。包括似然比检验在内的常见检验都不知道小样本性质（无偏性、幂函数的单调性、可接受性等）。我们将探讨自举技术在多元分析推理问题中的应用。经典的多元统计程序以线性、独立和正态假设为特征。这些程序在整个科学领域广泛用于分析实验和观测数据。在这些多变量分析中使用的一些估计的统计程序是粗糙的，因此，改进它们可能会对科学家得出的关于测量之间关系的结论产生重大影响。同样在这种情况下，我们目前关于统计决策程序性能的知识严格适用于重复大量次的实验。由于重复实验通常是昂贵的，所以通常以低频率进行。当重复数量较少时，对统计决策程序性能的研究有望为科学界的广泛领域提供有价值的信息和见解。

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.