Mathematical Sciences: Flowgraph and Saddlepoint Methods for Statistics

本研究涉及将流图和鞍点方法应用于统计问题，特别强调随机网络中的预测。随机网络模型是目前统计学研究的热点，可用于研究各种自然现象。考虑到诸如肾衰竭、癌症或艾滋病等疾病的进展。人们感兴趣的是对病人生存时间的预测。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首次通过时间，因此对患者生存时间的预测涉及对复杂随机网络的分析。本研究就是关于这样的预测。在涉及大量协变量和大量审查数据的情况下，开发了随机网络的贝叶斯预测分布的计算方法。这些方法与广义线性模型和比例风险模型结合使用，以便计算疾病任意两种状态之间首次传播时间的预测分布以及预测风险和预测生存函数，或者更一般地说，计算随机网络。这是一门对流程图的研究，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发展起来的，用于设计和分析复杂系统。例如，这些系统可以是制造过程的描述，人工器官的可靠性，或建筑项目的预计完成时间。传统的流程图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。本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，可为这些问题提供快速准确的近似。这里开发的结果适用于可靠性，工业，电气和系统工程领域，除了生存分析。＊＊＊

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. ***