Saddlepoint and Bootstrap Methods in Stochastic Systems and Related Fields

摘要 作者提出了一个完整的框架，实现随机系统中的非参数统计推断。这些随机系统是半马尔可夫过程，包括可靠性，多状态生存分析，流行病建模，通信和制造系统中最常用的随机模型。需要三个工具来完成框架：变换的余因子规则，反转变换的鞍点近似，以及与前两个工具一起提供统计推断的引导。如果缺少这三种工具中的任何一种，统计推断一般都不再可能。这样一个完整的理论是目标的控制论运动在40年代末至70年代中期，投入了大量的努力，以发展一个拉普拉斯变换方法，这样的建模。最终，这种方法失败了，因为很难反演所涉及的变换，一个非常成功的任务，通过使用鞍点近似。它也缺乏一个统计理论的推论，这是一个需要填补令人钦佩的自助。该提案的工作朝着实现控制论运动的最终目标迈出了一步：促进概率计算和随机系统的非参数（自举）推理，这是通过其他手段无法轻松实现的。没有鞍点辅助的自举推理模拟对于即使是中等大小和复杂度的系统也超出了计算可行性。本计画提出发展一个完整的架构，以实现复杂随机系统的非参数统计推论。这些随机系统包括大多数常用的随机模型，用于可靠性，多状态生存分析，流行病建模，通信和制造系统。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。在群体遗传学中，为处理自然选择、突变和遗传漂移的统计推断问题提供了解决方案;在海洋和电气工程中，为用于海面和信号处理的模型中的波峰高度分布提供了精确的近似值;在通过神经系统传递疼痛的生物模型中，给出了方法以允许关于驱动这些离子通道模型的波动极性的潜在机制的推论，其最终目的是帮助揭示控制痛觉的机制。

Abstract The author proposes to develop a complete framework for implementing nonparametric statistical inference in stochastic systems. These stochastic systems are semi-Markov processes and 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 for transforms, saddlepoint approximations to invert the transforms, and the bootstrap to provide statistical inference in conjunction with the two previous tools. With any one of the three tools missing, statistical inference is no longer generally possible. Such a complete theory was the goal of the cybernetics movement during the late 40s to mid 70s which devoted a great deal of effort into developing a Laplace transform approach to such modelling. Ultimately this approach failed due to the difficulty of inverting the transforms involved, a task very successfully performed by using saddlepoint approximations. It also lacked a statistical theory of inference, a need that is filled admirably by the bootstrap. The work of this proposal takes a step towards achieving the ultimate aims of the cybernetics movement: to facilitate probability computations and nonparametric (bootstrap) inference for stochastic systems that cannot be easily achieved by other means. Bootstrap simulation for inference without saddlepoint assistance is beyond computational feasibility for systems of even modest size and complexity. This project proposes to develop a complete framework for implementing nonparametric statistical inferencein complex stochastic systems. These stochastic systems include most of the commonly used stochastic modelsused in reliability, multi-state survival analysis, epidemic modelling, and communication and manufacturingsystems. The proposal also addresses significant questions in other disciplines where answers are lacking due to certain computational difficulties. In population genetics, solutions are provided for statistical inference problems dealing with natural selection, mutation and genetic drift; in ocean and electrical engineering accurate approximations are given for distributions of wave crest heights in models used for sea surfaces and in signal processing; in biological models for the transmission of pain through the nervous system, methods are given to allow inferences about the underlying mechanisms that drive the fluctuating polarities of these ion channel models with the ultimate aim of helping to reveal the mechanisms that control pain sensation.