Saddlepoint and Bootstrap Methods in Stochastic Systems and Related Fields

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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.