Mathematical Sciences: Multi-dimensional Statistical Analysis

经过深入研究的大量单变量分布有时被开发用于数据的统计分析；其他分布源于自然概率或物理考虑、统计性质或作为限制分布。大多数众所周知的分布都以一种以上的方式出现。虽然单变量分布提供了关于单个测量的信息，但它们不能促进测量之间关系的研究。在物理、生物和经济科学中的许多情况下，正是这些关系是人们感兴趣的，因此需要多变量分布。这项建议的目的是帮助理解著名的单变量理论的多变量版本。这一主题特别及时，因为最近计算和计算机图形能力的增加使多变量分布的使用变得更加广泛实用。多变量模型现在尤其被理解为将许多多变量家庭结合成凝聚力单位的基础结构。多变量分布的有意义的派生可以阐明在不同的应用中使用哪些模型。这项研究的一个结果是，可用于描述多变量数据的灵活模型将变得可用。

The large number of univariate distributions that have been thoroughly studied sometimes have been developed for use in statistical analyses of data; other distribution arise from natural probabilistic or physical considerations, from statistical properties or as limiting distributions. Most of the well-known distributions arise in more than one way. Although univariate distributions provide information about individual measurements, they cannot facilitate studies of relationships between measurements. In many situations in the physical, biological, economic sciences it is just these relationships that are of interest, for which multivariate distributions are required. The goal of this proposal is to contribute to the understanding of multivariate versions of the well-known univariate theory. This subject is particularly timely because recent increases in computational and computer graphics capabilities have made the use of multivariate distributions much more broadly practical. Multivariate models are now particularly understood in terms of underlying structures that unite many multivariate families into cohesive units. Meaningful derivations of multivariate distributions can shed light on which models to use in different applications. A consequence of this research is that flexible models useful in describing multivariate data will become available.