Saddlepoint and Bootstrap Accuracy with Applications to General Systems Theory

该项目有两个主要目标：(a)为鞍点近似的显著准确性找到解释，(b)继续开发在随机系统中实现非参数统计推断的框架。考虑目标(a)使用了两种数学工具：在解析数论中更常用来证明素数定理的Ikehara-Weiner定理，以及应用于矩生成函数（MGFs）反演公式的复积分方法。两种方法都着重于MGFs的解析延拓及其收敛带外的性质。这两种工具将用于简化和扩展已知的关于鞍点近似的均匀相对精度。目标(b)继续之前的工作，开发一个框架，用于在有限状态半马尔可夫过程的随机模型中实现自举推理。这些模型包括可靠性、多状态生存分析、流行病建模以及通信和制造系统中大多数常用的随机模型。完成这个框架需要三个工具：指定性能特征的拉普拉斯变换的协因子规则、反转这些变换的鞍点近似，以及与前面两个工具一起提供统计推断的bootstrap。现代统计方法使用的模型涉及复杂的分布，从中计算概率可能是一项艰巨的任务。这个任务通常通过使用鞍点近似来简化。这种近似通常只需要很少的努力就可以提供概率，并且通常可以达到2-3位有效数字的精度。对于这种惊人的准确性，研究人员仍然无法解释。本提案的(a)部分概述了研究者将考虑解释这种准确性的两种新方法。在需要从复杂分布中进行概率计算的现代方法中，研究者在提案的第(b)部分中考虑了程序。这项工作涉及在复杂随机系统中实现非参数统计推理的框架的发展，其中一些开始于控制论运动期间在工程中制定的复杂系统。这些随机系统包括大多数用于可靠性、多状态生存分析、流行病建模以及通信和制造系统的标准随机模型。在一般随机系统模型的背景下，目前还没有这样的通用方法来实现统计推断，因此研究者提出的框架将提供目前不可用的工具。该提案还解决了其他学科中由于某些计算困难而缺乏答案的重要问题。在海洋和电气工程中，给出了波涛汹涌时船体极端应力分布和信号处理中极端响应分布的精确近似；在量子物理学中，对规范函数提出了近似，规范函数的计算是量子理论的基础，即使在最简单的情况下，其计算也是困难的。

This project has two major goals: (a) to find explanations for the remarkable accuracy of saddlepoint approximations, and (b) to continue development of a framework for implementing nonparametric statistical inference in stochastic systems. Consideration of objective (a) uses two mathematical tools: the Ikehara-Weiner theorem, more commonly used in analytic number theory to prove the prime number theorem, and complex integration methods applied to inversion formulas of moment generating functions (MGFs). Both methods focus on the analytic continuation of MGFs and their properties outside of the convergence strip. These two tools will be used to streamline and extend what is known concerning the uniformly relative accuracy of saddlepoint approximations. Objective (b) continues previous work on the development of a framework for implementing bootstrap inference in stochastic models that are finite-state semi-Markov processes. These models include most of the commonly used stochastic models in reliability, multi-state survival analysis, epidemic modeling, and communication and manufacturing systems. Three tools are required to complete the framework: cofactor rules specifying the Laplace transforms for performance characteristics, saddlepoint approximations to invert these transforms, and the bootstrap to provide statistical inference in conjunction with the two previous tools. Modern statistical methods use models that involve complicated distributions from which the computation of probabilities can be a formidable task. This task is often simplified by using saddlepoint approximations. Such approximations generally provide probabilities with very little effort and most often achieve 2-3 significant digit accuracy. Explanations for this remarkable accuracy have continued to elude researchers. Part (a) of this proposal outlines two new approaches the investigator will consider to explain this accuracy. Among the modern methods that require probability computations from complicated distributions are the procedures the investigator considers in part (b) of the proposal. This work concerns the development of a framework for implementing nonparametric statistical inference in complex stochastic systems some of which began with the complex systems formulated in engineering during the cybernetics movement. These stochastic systems include most of the standard stochastic models used in reliability, multi-state survival analysis, epidemic modeling, and communication and manufacturing systems. No such general methodology currently exists for implementing statistical inference in the context of general stochastic systems models so the framework proposed by the investigator would provide tools that are currently unavailable. The proposal also addresses significant questions in other disciplines where answers are lacking due to certain computational difficulties. In ocean and electrical engineering accurate approximations are given for distributions of extreme hull stress during heavy seas and distributions for extreme responses in signal processing; in quantum physics, approximations are proposed for "gauge" functions, the computation of which are fundamental in quantum theory, and whose computation is difficult even in the simplest cases.