Saddlepoint Methods in Statistics

9970785第一个项目有三部分。(1)对许多矩阵参数特殊函数，包括重要的贝塞尔函数和超几何函数，提出了拉普拉斯近似。这些函数出现在方差分析设置中，并确定大多数测试统计量以及样本特征值的非中心分布。通常是非中心分布的矩生成函数（mgf）用特殊函数表示；在这种情况下，MGF可以近似，然后使用鞍点近似反转。(2)鞍点近似用于计算复杂随机反馈系统的可靠性和故障率。系统应用包括随机漫步、队列、多状态生存模型和一些随机流行病模型中的各种通过时间。这些方法可以非常精确地确定系统的瞬态行为，包括系统设计中有用的许多重要性能特征。(3)鞍点法是近似所有这些分布和密度的稳定律。这一发展将使在金融数学中使用稳定误差律实现贝叶斯计算变得容易。第二个项目由三个部分组成。(1)对某些类型的特殊函数提出了非常精确的近似。这些特殊的函数出现在物理科学的所有领域，由数学家创造，因为它们作为重要科学问题的解决方案反复出现，也因为它们极难计算。在统计学领域，这些函数决定了多变量分析中许多常用统计检验的效力和性能。即使在现代计算环境中，这些函数的计算也需要大量的计算时间。所提出的近似是简单的，明确的，高度精确的，并且应该在所有物理和工程科学中有用。（二）对各种排队系统、生存模型和随机流行病模型等复杂随机系统的性能特征给出了一般逼近方法。这种随机系统和网络是所有科学领域的基础，包括计算机系统网络、生态系统、制造业生产管理方法和可靠性测试的模型。项目的第二部分开发了评估这些性能特征的方法，这些方法反过来又允许对系统设计进行实际考虑。（三）提出了金融数学中统计推理的一些方法。在这个领域，某些概率分布，被称为稳定分布，在金融回报建模中非常有用。不幸的是，这些分布仍然非常难以计算，现有的计算例程也不是很精确。这个建议提出了一些非常简单和高度精确的近似，应该使这种分析例行和容易。

9970785The first project has three parts. (I) Laplace-type approximations are developed for many of the matrix argument special functions including the important Bessel and hypergeometric functions. These functions arise in MANOVA settings and determine the noncentral distributions for the majority of test statistics as well as the sample eigenvalues. Often it is the moment generating function (mgf) of the noncentral distribution that is specified in terms of the special function; in this case the mgf can be approximated and then inverted using a saddlepoint approximation. (II) Saddlepoint approximations are used in computing reliabilities and failure rates of complicated stochastic feedback systems. System applications include various passage times in random walks, queues, multistate survival models, and some stochastic epidemic models. The methods allow for very accurate determination of the transient behavior of the systems including many important performance characteristics useful in system design. (III) Saddlepoint methods are suggested for the stable laws to approximate all of these distributions and densities. This development should make it easy to implement Bayesian computation using stable error laws as needed in financial mathematics.The second project consists of three parts. (I) Very accurate approximations are proposed for some classes of special functions. These special functions arise in all areas of the physical sciences and were created by mathematicians because they repeatedly arise as the solutions to important scientific problems and also because they are extremely difficult to compute. In the field of statistics, these functions determine the power and performance of many commonly used statistical tests in multivariate analysis. Computation of these functions requires immense amounts of computing time even in our modern computing environment. The proposed approximations are simple, explicit, highly accurate, and should be useful in all the physical and engineering sciences. (II) General methods of approximation are given for the performance characteristics of complicated stochastic systems including various queuing systems, survival models, and stochastic epidemic models. Such stochastic systems and networks underlie all areas of science and include models for computer system networks, ecosystems, production management methods in manufacturing, and reliability testing. This second portion of the project develops methods for assessing these performance characteristics which, in turn, allow for the practical consideration of system design. (III) Some methods for statistical inference in financial mathematics are proposed. In this area, certain probability distributions, referred to as the stable distributions, are known to be very useful in modeling financial returns. Unfortunately these distributions are still extremely difficult to compute and existing routines for their computation are not very accurate. This proposal suggests some very simple and highly accurate approximations that should make such analysis routine and easy.