Saddlepoint and Bootstrap Accuracy with Applications to General Systems Theory

鞍点和自举精度及其在一般系统理论中的应用

• 批准号: 1104474
• 负责人: Ronald Butler
• 金额: $ 15.44万
• 依托单位: Southern Methodist University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104474&HistoricalAwards=false
• 关键词: Saddlepoint; Bootstrap; Accuracy; Applications; General

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.

该项目有两个主要目标：（a）寻找鞍点近似的显着准确性的解释，（B）继续发展的框架，实现随机系统中的非参数统计推断。目标（a）的考虑使用了两种数学工具：Ikehara-Weiner定理，在分析数论中更常用于证明素数定理，以及应用于矩生成函数（MGF）的反演公式的复积分方法。这两种方法都侧重于MGF的解析延拓及其在收敛带之外的性质。这两个工具将用于简化和扩展已知的关于鞍点近似的一致相对精度。目标（B）继续之前的工作，开发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的框架。这些模型包括大多数常用的随机模型在可靠性，多状态生存分析，流行病建模，通信和制造系统。需要三个工具来完成框架：余因子规则指定的拉普拉斯变换的性能特征，鞍点近似反转这些变换，并引导提供统计推断与前两个工具。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。这一任务通常通过使用鞍点近似来简化。这样的近似值通常以非常小的努力提供概率，并且最经常达到2-3个有效数字的准确度。对于这种惊人的准确性，研究人员一直无法解释。本提案的（a）部分概述了研究人员将考虑的两种新方法，以解释这种准确性。在现代方法中，需要从复杂的分布中计算概率的是研究者在建议的（B）部分中考虑的程序。这项工作涉及的发展框架实施非参数统计推断在复杂的随机系统，其中一些开始与复杂的系统制定的工程控制论运动期间。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。目前还没有这样的一般方法来实现统计推断的背景下，一般的随机系统模型，所以由调查员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，精确的近似值被用于计算在大风浪中船体的极端应力分布和信号处理中的极端响应分布;在量子物理学中，近似值被用于“规范”函数，其计算是量子理论的基础，即使在最简单的情况下也很难计算。