Mathematical Sciences: Flowgraph and Saddlepoint Methods for Statistics

DMS 9625672 Huzurbazar这项研究涉及到流图和鞍点方法在统计学问题中的应用，特别强调随机网络中的预测。随机网络模型是当前统计学领域的研究热点，可用于研究各种自然现象。想想肾衰竭、癌症或艾滋病等疾病的进展情况。有趣的是预测病人的生存时间。生存时间可以被认为是随机网络中从一种状态到另一种状态的第一次通过时间，因此对患者生存时间的预测涉及对复杂随机网络的分析。这项研究就是关于这样的预测的。提出了一种在包含大量协变量和严格删失数据的情况下计算随机网络的贝叶斯预测分布的方法。这些方法与广义线性模型和比例风险模型结合使用，从而计算疾病任何两个状态之间的首次通过时间的预测分布以及预测风险和预测生存函数，或者更一般地，随机网络。这是一项对流程图的研究，它超越了生存分析的自然重点领域，扩展到工程系统的几个不同领域。流程图最初是在工程科学中开发的，用于设计和分析复杂的系统。例如，这些系统可以是对制造过程、人造器官的可靠性或建筑项目预计完工时间的描述。传统上，流程图的分析一直受到计算困难的阻碍。目前的工程方法涉及耗时的计算机模拟。这项研究的计算方面是基于鞍点近似。鞍点近似是一种高性能的计算密集型技术，它为这些问题提供了快速而准确的近似。这里开发的结果除了适用于生存分析外，还适用于可靠性、工业、电气和系统工程等领域。***

DMS 9625672 Huzurbazar This research involves the application of flowgraph and saddlepoint methods to problems in statistics with particular emphasis on prediction in stochastic networks. Stochastic network models are of current interest in statistics and can be applied to study a variety of natural phenomena. Consider the progression of diseases such as kidney failure, cancer, or AIDS. Of interest is the prediction of a survival time for a patient. Survival times can be thought of as first passage times from one state to another in a stochastic network so that prediction of a survival time for a patient involves analysis of a complex stochastic network. This research is concerned with such prediction. Methodology is developed for computation of Bayesian predictive distributions for stochastic networks in situations involving a multitude of covariates and heavily censored data. These methods are used in conjunction with generalized linear models and proportional hazards models, so that predictive distributions as well as predictive hazards and predictive survival functions for first passage times between any two states of a disease, or more generally, stochastic networks, are computed. %%% This is a study of flowgraphs which extends beyond the natural emphasis area of survival analysis into several diverse areas of engineering systems. Flowgraphs were originally developed in the engineering sciences to design and analyze complex systems. For example, these systems could be descriptions of a manufacturing process, the reliability of an artificial organ, or the predicted time to completion of a building project. The analysis of flowgraphs traditionally has been hampered by computational difficulties. Current engineering methods involve time-consuming computer simulations. The computational aspects of this research are based on saddlepoint approximations. Saddlepoint approximations are high performance computationally intensive techniques that provid e fast and accurate approximations to these problems. The results developed here are applicable in the areas of reliability, and industrial, electrical, and systems engineering, in addition to survival analysis. ***