Saddlepoint and Bootstrap Methods in Stochastic Systems and Related Fields

摘要作者建议建立一个在随机系统中实现非参数统计推断的完整框架。这些随机系统是半马尔可夫过程，包含了可靠性、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：用于转换的余因数规则、用于反转转换的鞍点近似以及用于与前两个工具一起提供统计推断的引导。由于这三种工具中的任何一种都缺失了，统计推断通常不再可能。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，该运动投入了大量的努力来开发拉普拉斯变换方法来进行这种建模。最终，这种方法失败了，因为很难对所涉及的变换进行倒置，这项任务通过使用鞍点近似非常成功地完成了。它也缺乏推论的统计理论，这一需求被自助法很好地填补了。这项建议的工作朝着实现控制论运动的最终目标迈出了一步：促进随机系统的概率计算和非参数(自举)推理，这是其他方法无法轻松实现的。对于即使是中等规模和复杂的系统，在没有鞍点辅助的情况下进行推理的Bootstrap模拟在计算上是不可行的。这个项目建议开发一个在复杂随机系统中实现非参数统计推断的完整框架。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常用的大多数随机模型。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。在群体遗传学中，提供了涉及自然选择、突变和遗传漂移的统计推断问题的解决方案；在海洋和电气工程中，给出了用于海面和信号处理的模型中波峰高度分布的精确近似；在用于通过神经系统传递疼痛的生物模型中，给出了允许对驱动这些离子通道模型波动极性的潜在机制的推断的方法，最终目的是帮助揭示控制痛感的机制。

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.