Mathematical Sciences: Models and Structures of Multivariate Distribution

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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.