Mathematical Methods for Approximately Exact Statistical Inference

近似精确统计推断的数学方法

• 批准号: 0906569
• 负责人: John Kolassa
• 金额: $ 12.26万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906569&HistoricalAwards=false
• 关键词: Mathematical; Methods; Approximately; Exact; Statistical

This proposed research applies a variety of mathematical techniques, including multivariate complex analysis and combinatorics, to open questions concerning inference from small samples. This inference includes standard frequentist techniques including the calculation of p-values and confidence intervals. The following aims are undertaken: 1. The approximation of DiCiccio and Martin (1993), conditional p-values using an approximate equivalence relation between a Bayesian credible region and a frequentist critical region, are improved, by determining an optimal or near-optimal set of initial conditions for the partial differential equation that is involved in the approximation. 2. Exact-enumeration techniques are applied to conditional inference in Cox regression. 3. An asymptotic approximation are constructed for the conditional distribution of a likelihood ratio statistic, under the regularity conditions far weaker then generally present in the existent literature.Experimentalists routinely ask how well their data fits a hypothesis about the mechanism generating their data; the truth or falsehood of this hypothesis routinely is of importance to society as a whole. For example, in a medical clinical trial, one might investigate how closely data conform to a hypothesis that a new drug is equivalent at treating a particular disease to a standard drug, and in an engineering study, one might investigate how closely the data conform to a hypothesis that a part designed in a new way lasts no longer than a part designed in a conventional way. Disproving such a hypothesis often leads to adopting an alternative hypothesis that a new drug or part design actually represents an improvement. Disproving such a hypothesis involves a probabilistic proof by contradiction, in which investigators calculate the probability of observing data representing evidence at least as strong against the initial hypothesis, and reject this hypothesis if this probability is small. For example, investigators interested in a new design for a low-emission vehicle might compare a new battery design to an old design, in an experiment in which batteries of both types are installed in vehicles and tested under a variety of conditions. In situations like this, the method for quantifying evidence against the hypothesis of equal reliability of both batteries is well established. This research helps to attach probabilities to the various possible outcomes of the experiment under various assumptions about the relative reliabilities of the batteries, accounting for variation in experimental conditions and for various non-battery reasons for vehicle failure. Similar questions arise in medicine, finance, the social sciences, and many other fields of interest.

本研究应用多种数学技术，包括多元复分析和组合学，来解决有关小样本推断的开放性问题。这种推断包括标准的频率学技术，包括p值和置信区间的计算。执行的目标如下：DiCiccio和Martin（1993）的近似，使用贝叶斯可信区域和频率临界区域之间的近似等价关系的条件p值，通过确定近似中涉及的偏微分方程的最优或近最优初始条件集来改进。2. 精确枚举技术应用于Cox回归中的条件推理。3. 本文构造了似然比统计量的条件分布的渐近逼近，其正则性条件远弱于现有文献。实验学家通常会问，他们的数据与产生数据的机制的假设是否吻合；这种假设的真假通常对整个社会都很重要。例如，在医学临床试验中，人们可能会调查数据在多大程度上符合一个假设，即一种新药在治疗特定疾病方面与标准药物等效；在一项工程研究中，人们可能会调查数据在多大程度上符合一个假设，即用新方法设计的部分不会比用传统方法设计的部分持续时间更长。反驳这样的假设通常会导致采用另一种假设，即新药或部分设计实际上代表了一种改进。反驳这样一个假设涉及到反证法的概率证明，在这个过程中，研究人员计算观察到的数据至少与最初的假设有同样强的证据的概率，如果这个概率很小，就拒绝这个假设。例如，对低排放汽车的新设计感兴趣的研究人员可能会将新设计的电池与旧设计的电池进行比较，在一项实验中，两种类型的电池都安装在汽车上，并在各种条件下进行测试。在这种情况下，量化证据反对两种电池可靠性相等的假设的方法已经建立起来了。这项研究有助于在关于电池相对可靠性的各种假设下，为实验的各种可能结果附加概率，考虑到实验条件的变化以及车辆故障的各种非电池原因。类似的问题也出现在医学、金融、社会科学和许多其他感兴趣的领域。