Mathematical Sciences: Saddlepoint Methods in Statistics

本研究的第一部分涉及在预测密度不具有可处理形式的情况下贝叶斯预测密度的计算。这种设置发生在随机网络模型中，其中可以使用流程图方法来推导其矩生成函数。然后使用鞍点方法将这些生成函数转换为密度和累积生成函数。后验分布上的后续混合可以用模拟或拉普拉斯解析方法来完成。我们还开发了多变量分析中p值计算的重要抽样方法。实际上，本主题领域的所有常见测试都可以使用我们将开发的模拟方案进行计算，并且在工作站上的运行时间应少于五分钟。还要考虑处理非线性随机系统响应近似的附加工作。系统理论包括生产管理的研究，其中产品沿着装配线的流动是感兴趣的，排队理论，电网的可靠性，高速公路系统的交通流量等。当这些系统受到随机性影响时，对它们的数学研究是基于一个叫做变换理论的数学领域。到目前为止，转换理论还没有被证明是一个特别有用的工具，对于更实际的工程师来说，它可以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。这项研究关注的是应用新的数学工具，称为鞍点近似来解决这些问题，这将允许从变换理论中提取实际的答案。一个随机系统现在可能需要10个小时的计算机模拟来分析，当这些鞍点近似的全部功率与现有的转换理论的功率相结合时，在计算机上只需要几分钟就可以分析。本研究涉及使用鞍点近似来分析系统，并提取有关此类系统的实际答案。涉及多个变量的统计分析通常要求使用多变量方法分析数据。在这种情况下，测试各种假设的显著性是不容易完成的，因为所达到的显著性水平的计算是不精确的，需要近似。提出的研究提供了特殊的计算机模拟方法，这将导致对大多数多变量检验达到显著性水平的几乎精确的计算。

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.