Mathematical Sciences: Saddlepoint Methods in Statistics

这项研究的第一部分涉及在预测密度没有易于处理的形式的情况下的贝叶斯预测密度的计算。这样的设置发生在随机网络模型中，其中可以使用流图方法来推导它们的矩生成函数。然后使用鞍点方法将这些母函数转换为密度和累积量母函数。后验分布上的后续混合是用模拟或拉普拉斯的分析方法来完成的。我们还发展了用于多元分析中p值计算的重要抽样方法。使用我们将开发的模拟方案，这一主题领域中的几乎所有常见测试都将是可计算的，并且在工作站上的运行时间应少于5分钟。还需要考虑处理非线性随机系统中响应的近似的附加工作。系统理论包括生产管理研究、排队论、电力网络的可靠性、公路系统上的交通流量等等。当这些系统受到随机性的影响时，这些系统的数学研究是基于一个被称为变换理论的数学领域。到目前为止，对于更实际的工程师来说，变换理论还没有被证明是一个特别有用的工具，用于提取关于系统的实际答案。工程师们大多倾向于使用非常耗时的计算机模拟来进行这种分析。这项研究致力于将被称为鞍点近似的新数学工具应用于这些问题，从而从变换理论中提取出实际的答案。在随机系统现在可能需要10个小时的计算机模拟来分析的情况下，当这些鞍点近似的全部能力与现有的变换理论的能力相结合时，在计算机上分析应该只需要几分钟的时间。这项研究致力于使用鞍点近似来分析系统，并提取关于这类系统的实际答案。涉及一个以上变量的统计分析通常需要使用多变量方法来分析数据。在这种情况下，检验各种假设的重要性并非易事，因为所获得的重要性水平的计算并不准确，需要近似。这项拟议的研究提供了特殊的计算机模拟方法，将导致对大多数多变量测试达到的显著水平进行几乎准确的计算。

The first portion of this research is concerned with the computation of Bayesian predictive densities in settings where the predictive density does not have a tractable form. Such settings occur in stochastic network models where flowgraph methods can be used to derive their moment generating functions. Saddlepoint methods are then used to convert these generating functions into densities and cumulant generating functions. Subsequent mixing over the posterior distribution is accomplished with either simulation or the analytic method of Laplace. We also develop importance sampling methods for p-value computation in multivariate analysis. Virtually all the common tests in this subject area will be computable with the simulation schemes we shall develop and the run times should amount to less than five minutes on a work station. Additional work dealing with approximation of responses in non-linear stochastic systems is to be considered. Systems theory encompasses such things as the study of production management, where the flow of products along an assembly line is of interest, queuing theory, reliability of electrical networks, flow of traffic on a highway system, etc. The mathematical study of these systems when they are subject to randomness is based on an area of mathematics called transform theory. So far transform theory has not proven to be a particularly useful tool to the more practical engineer for extracting practical answers about systems. Engineers are mostly inclined to use very time-consuming computer simulations for this analysis. This research is concerned with applying new mathematical tools called saddlepoint approximations to these problems that will allow practical answers to be extracted from the transform theory. Where a random system might now take 10 hours of computer simulation to analyze, it should take only minutes on the computer to analyze when the full power of these saddlepoint approximations are combined with the existing pow er of transform theory. This research is concerned with using saddlepoint approximations to analyze systems and for extracting practical answers about such systems. Statistical analyses that involve more than one variable often require that the data are analyzed using multivariate methods. In this setting testing the significance of various hypotheses is not easily accomplished since the computation of attained levels of significance is not exact and requires approximation. The proposed research offers special computer simulation methods that will lead to virtually exact computation of attained significance levels for the majority of multivariate tests.