Mathematical Methods for Approximately Exact Statistical Inference

这项研究应用了多种数学技术，包括多元复分析和组合学，以解决有关小样本推断的问题。 该推断包括标准频率论技术，包括p值和置信区间的计算。 主要目的如下：1. DiCiccio和Martin（1993）的近似，条件p值使用贝叶斯可信区域和频率论临界区域之间的近似等价关系，通过确定近似中涉及的偏微分方程的最佳或接近最佳的初始条件集来改进。2.将精确枚举技术应用于考克斯回归中的条件推理。 3. 在比现有文献中普遍存在的规则性条件弱得多的条件下，为似然比统计量的条件分布构造了一个渐近近似。实验主义者经常询问他们的数据与关于产生他们的数据的机制的假设的吻合程度;这个假设的真实性或虚假性经常对整个社会具有重要意义。 例如，在医学临床试验中，人们可能会调查数据与一种新药在治疗特定疾病方面等同于标准药物的假设的符合程度，而在工程研究中，人们可能会调查数据与一种假设的符合程度，即以新方式设计的部件不会比以传统方式设计的部件持续更长时间。 反驳这样的假设往往会导致采用另一种假设，即新药或部件设计实际上代表了一种改进。 反驳这样一个假设涉及一个矛盾的概率证明，其中调查人员计算观察到的数据代表的证据至少与初始假设一样强的概率，如果这个概率很小，就拒绝这个假设。 例如，对低排放车辆的新设计感兴趣的研究人员可能会将新电池设计与旧设计进行比较，在实验中，两种类型的电池都安装在车辆中并在各种条件下进行测试。在这种情况下，针对两种电池的同等可靠性的假设，量化证据的方法已经得到了很好的建立。这项研究有助于附加概率的各种可能的结果的实验在各种假设的相对可靠性的电池，占变化的实验条件和各种非电池的原因车辆故障。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。

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.