Mathematical Sciences: Models and Structures of Multivariate Distribution

DMS 9626339马歇尔 本研究从模型选择的角度研究寿命分布。 一个阶段涉及在族中生成参数的方法。 给定一个具有分布函数和生存函数的随机变量，有许多方法可以施加参数，在某些情况下通过取随机变量的函数，在某些情况下通过取分布函数或生存函数。 以下是一些例子：（1）位置和尺度参数;（2）脆弱性和恢复力参数;（3）功率参数;（4）卷积参数（用于特殊族，如伽马族）;（5）几何极端稳定参数。 研究人员研究这些参数族的顺序关系。 例如，对于哪些参数族，危险函数是有序的;对于哪些族，存在随机或凸序。 参数及其特性和它们的排序的引入是关于如何将结果扩展到多元分布的洞察力的基础。 虽然正态分布在许多应用和统计理论中起着核心作用，但它不能作为工业应用中产生的寿命数据（如制造数据，材料数据，可靠性数据）的模型，也不能作为健康或环境科学中的生存分析或风险分析的模型。 本研究的重点是如何生成非正态参数族，这将提供一个洞察行为模式和预测这样的数据。 特别是，正在开发的方法有助于提升工程应用中的风险和可靠性分析。 在这项研究的第二阶段是多变量版本的发展，考虑到测量之间的依赖关系。

DMS 9626339 Marshall This research involves the study of life distributions from the point of view of model selection. One phase involves methods for generating parameters in families. Given a random variable with distribution and survival functions, there are a number of ways to impose parameters, in some instances by taking a function of the random variable, and in some cases by taking a function of the distribution or survival function. The following are some examples: (1) Location and scale parameters; (2) Frailty and resilience parameters; (3) Power parameter; (4) Convolution parameter (for special families such as the gamma family); (5) Geometric-extreme stable parameter. The investigators study order relations for these parametric families. For example, for which parametric families are the hazard functions ordered; for which families is there a stochastic or convex ordering. The introduction of parameters and their characteristics and their orderings is fundamental to an insight concerning how to extend the results to multivariate distributions. Although the normal distribution plays a central role in many applications and in statistical theory, it does not serve as a model for lifetime data arising in industrial applications such as manufacturing data, materials data, reliability data, nor does it serve as a model for survival analysis or risk analysis in the health or environmental sciences. This research focuses on how to generate non-normal parametric families that will provide an insight into behavior patterns and prediction for such data. In particular, the methods being developed serve to upgrade risk and reliability analyses in engineering applications. A second stage in this research is the development of multivariate versions that take account of dependencies among measurements.