Saddlepoint and Bootstrap Methods in Stochastic Systems and Related Fields

作者提出了一个完整的框架来实现随机系统中的非参数统计推理。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、通信和制造系统中大多数常用的随机模型。完成这个框架需要三个工具：用于变换的协因式规则、用于反转变换的鞍点近似，以及用于与前面两个工具结合提供统计推断的自举。如果缺少这三种工具中的任何一种，通常就不再可能进行统计推断。这样一个完整的理论是控制论运动在40年代末到70年代中期的目标，它投入了大量的精力来发展这种建模的拉普拉斯变换方法。最终，这种方法失败了，因为所涉及的变换反转困难，这是一个通过使用鞍点近似非常成功地完成的任务。它还缺乏推理的统计理论，而这一需求被bootstrap令人钦佩地填补了。这一建议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，这是其他方法无法轻易实现的。在没有鞍点辅助的情况下进行推理的自举模拟超出了中等规模和复杂性系统的计算可行性。本项目拟建立一个完整的框架，用于在复杂随机系统中实现非参数统计推理。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，提供了处理自然选择、突变和遗传漂变的统计推断问题的解决方案；在海洋和电气工程中，海面模型和信号处理中给出了波峰高度分布的精确近似；在疼痛通过神经系统传递的生物学模型中，给出了方法来推断驱动这些离子通道模型极性波动的潜在机制，最终目的是帮助揭示控制疼痛感觉的机制。

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.