Saddlepoint and Bootstrap Accuracy with Applications to General Systems Theory

这个项目有两个主要目标：(A)找到鞍点近似惊人精度的解释，(B)继续开发在随机系统中实现非参数统计推断的框架。对目标(A)的考虑使用了两个数学工具：池原-韦纳定理，这是解析数论中更常用的证明素数定理的方法，以及应用于矩母函数(MGFs)求逆公式的复积分方法。这两种方法都侧重于MGFs的解析延拓及其在收敛带外的性质。这两个工具将被用来简化和扩展关于鞍点近似的统一相对精度的已知内容。目标(B)继续先前关于在有限状态半马尔可夫过程的随机模型中实现Bootstrap推理的框架的开发。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常用的大多数随机模型。完成该框架需要三个工具：为性能特征指定拉普拉斯变换的余因数规则，用于反转这些变换的鞍点近似，以及用于结合前面两个工具提供统计推断的引导程序。现代统计方法使用的模型涉及复杂的分布，从这些模型计算概率可能是一项艰巨的任务。这项任务通常通过使用鞍点近似来简化。这样的近似通常只需很少的努力就可以提供概率，并且通常可以达到2-3个显著的数字精度。对于这种惊人的精确度，研究人员一直难以做出解释。本提案的(A)部分概述了调查员将考虑用来解释这一准确性的两种新方法。在需要根据复杂分布计算概率的现代方法中，有一种是调查人员在提案的(B)部分中考虑的程序。这项工作涉及到在复杂随机系统中实现非参数统计推断的框架的发展，其中一些开始于控制论运动期间在工程中形成的复杂系统。这些随机系统包括可靠性、多状态生存分析、流行病建模以及通信和制造系统中使用的大多数标准随机模型。目前尚不存在在一般随机系统模型范围内实施统计推断的一般方法，因此，调查员提出的框架将提供目前尚不具备的工具。该提案还解决了由于某些计算困难而缺乏答案的其他学科中的重要问题。在海洋和电气工程中，给出了在大浪中的极限船体应力分布和信号处理中的极限响应分布的精确近似；在量子物理中，提出了“规范”函数的近似，它的计算是量子理论的基础，即使在最简单的情况下，它的计算也是困难的。

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.