Mathematical Sciences: Multi-dimensional Statistical Analysis

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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.