Saddlepoint Methods in Statistics

统计中的鞍点方法

• 批准号: 9970785
• 负责人: Ronald Butler
• 金额: $ 12万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-15 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970785&HistoricalAwards=false
• 关键词: Saddlepoint; Methods; Statistics

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.

9970785第一个项目有三个部分。(I)拉普拉斯型近似是为许多矩阵参数的特殊功能，包括重要的贝塞尔函数和超几何函数。 这些函数出现在MANOVA设置中，并确定大多数检验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母函数（moment generating function，mgf）被指定为特殊函数;在这种情况下，mgf可以近似，然后使用鞍点近似进行反演。(II)鞍点近似用于计算复杂随机反馈系统的可靠度和失效率。 系统的应用包括各种通过时间的随机行走，排队，多态生存模型，和一些随机流行病模型。 该方法允许非常准确地确定系统的瞬态行为，包括在系统设计中有用的许多重要的性能特征。(III)鞍点方法建议稳定的法律，以近似所有这些分布和密度。 这一发展应该可以很容易地实现贝叶斯计算使用稳定的错误法律所需要的金融数学。第二个项目包括三个部分。(I)对某些特殊函数类提出了非常精确的近似。 这些特殊函数出现在物理科学的所有领域，并由数学家创造，因为它们反复出现作为重要科学问题的解决方案，也因为它们非常难以计算。 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功效和性能。 即使在我们现代的计算环境中，这些函数的计算也需要大量的计算时间。 建议的近似是简单的，明确的，高精度的，应该是有用的，在所有的物理和工程科学。(II)给出了复杂随机系统性能特征的一般逼近方法，包括各种排队系统、生存模型和随机传染病模型。 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、制造业的生产管理方法和可靠性测试的模型。 该项目的第二部分开发了评估这些性能特征的方法，这些性能特征反过来又允许系统设计的实际考虑。(III)提出了金融数学中统计推断的一些方法。在这一领域，某些概率分布，称为稳定分布，在建模金融收益时非常有用。 不幸的是，这些分布仍然非常难以计算，并且现有的计算例程不是很准确。 这个建议提出了一些非常简单和高度精确的近似，应该使这种分析程序和容易。