Saddlepoint and Bootstrap Methods in Systems Theory and Survival Analysis

摘要：系统理论和生存分析中的鞍点和Bootstrap方法。随机系统是现代科学大多数领域中应用的模型的基础。这些概念已经发展了很长一段时间，但最近受到控制论运动的严重影响，控制论运动跨越了40年代末到70年代末。那些参与这一运动的电气工程方面的人使用流程图来表示系统。有趣的时变系统特性自然地用与流图相关的拉普拉斯变换来表征。他们的努力导致了有限状态半马尔可夫系统的一般理论，推广了今天仍然普遍应用的限制性马尔可夫系统。不幸的是，这种方法中转换的反转被证明是一个挑战，最终限制了流程图的影响。这个建议解决了这些反转以及其他缺失的工具，以便一般系统理论可以充分发展成一个完整的数学学科。这些反演工具使用鞍点方法，可以确定复杂系统的瞬态行为。此外，引入了基于系统观测的非参数统计推理的自举方法。在工程和生物医学科学中的应用范围包括通信和计算机网络，排队理论，多状态生存模型，以及与这些模型相关的竞争风险背景下的左/右和区间审查。更具体地说，本提案解决了以下问题：(1)需要新的技术来反演表征系统复杂行为的拉普拉斯变换；作者提出了几种新的鞍点方法，利用最陡下降法来实现这一目标。(2)描述系统特征的变换需要以使鞍点方法易于使用的方式来指定。通常使用梅森法则，但由于其形式，它太复杂，无法应用于大型系统。作者提出了几个可选的辅助因子规则，大大简化了鞍点反演的计算。(3)利用自举及其与经验变换的关系，对这类半马尔可夫系统进行非参数统计推理。单自举和双自举为计算与半马尔可夫过程相关的生存函数和风险函数的置信带提供了一种实用的方法。所提出的双引导是通过鞍点反转实现的。本文研究了具有反馈的动态随机系统的统计和概率特性。在现代科学的大多数领域，这种系统的模型通常用于进行推断和预测。例如，在工程中，通信或计算机系统从一种状态变化到另一种状态，这些状态转换的动力学决定了系统的演化。该方案考虑了这种演化动态系统的可靠性或性能评估的计算和估计。所提出的方法在可靠性分析、电气工程、生物医学和制造业中具有重要的应用。

AbstractPI: R. W. ButlerDMS-0202284Title: Saddlepoint and Bootstrap Methods in System Theory and Survival AnalysisStochastic systems underlie the models applied in most areas of modern science. These concepts have evolved over a long period of time but were heavily influenced more recently by the cybernetics movement spanning the late 40s until the late 70s. Those involved on the electrical engineering side of this movement used flow graphs to represent the systems. Interesting time-dependent system characteristics were naturally characterized in terms of the Laplace transforms that could be associated with the flow graph. Their efforts lead to a general theory of finite state semi-Markov systems that generalized the very restrictive Markov systems that are commonly applied even today. Unfortunately, the inversion of the transforms in this approach proved to be a challenge that ultimately limited the impact of the flow graphs. This proposal addresses these inversions along with other missing tools so that a general systems theory may be fully developed into a complete mathematical discipline. These inversion tools, using saddlepoint methods, allow for the determination of the transient behavior of complex systems. In addition, the bootstrap is introduced for nonparametric statistical inference about the true system based on system observation. The range of applications in the engineering and biomedical sciences include communication and computer networks, queueing theory, multi-state survival models, and right/left and interval censoring in the context of the competing risks associated with such models. More specifically, this proposal addresses the following issues: (1) New techniques are required for inverting the Laplace transforms that characterize the complex behavior of systems; the author proposes several new saddlepoint methods for achieving this by using the method of steepest descents. (2) Transforms describing system characteristics need to be specified in ways that make the saddlepoint methods easy to use. Often Mason's rule is used but, because of its form, it is much too complicated to apply to large systems. The author has proposed several alternative co-factor rules that greatly simplify computations for the saddlepoint inversions. (3) Nonparametric statistical inference is to be developed for such semi-Markov systems using the bootstrap and its relationship to empirical transforms. The single and double bootstrap lead to a practical means for computing confidence bands of survival and hazard functions related to the semi-Markov process. The proposed double bootstrap is implemented through saddlepoint inversions.This proposal studies the statistical and probabilistic traits of dynamic stochastic systems with feedback. The models for such systems are commonly used to make extrapolations and predictions in most areas of modern science. In engineering, for example, a communication or computer system changes from state to state and the dynamics of these state transitions determine the evolution of the system. This proposal considers the computation and estimation of reliability or performance evaluation for such an evolving dynamic system. The proposed methods have important applications in reliability analysis, electrical engineering, biomedical sciences and manufacturing.