Generalized Fiducial Inference for Modern Statistical Problems

在这项提案中，研究人员带着一系列现代问题重新审视了费舍尔备受争议的基准论点。这是由徐和魏拉汉迪()提出的广义推理的成功所推动的，这实际上导致了与费舍尔的信义推理(Hannig，Iyer和Patterson，2006)相同的结果。研究人员并不试图推导出一个新的“无悖论的信义推理理论”。取而代之的是，在最小的假设下，研究人员提出了一个新的简单的基准配方，可以通过构造广义基准分布来进行统计推断。这个公式的灵感来自广义枢轴量的概念，设计成在许多实际应用中都很容易应用。它可以应用于参数空间的任何维度(即，包括非参数问题)，并且它经常导致统计过程是渐近精确的，更重要的是，具有非常好的近似小样本性质。研究人员建议研究统计问题的广义基准线分布的理论性质，并将他们的发现应用于更广泛感兴趣的各种问题。系统地研究广义基准推理的性质将增加我们对统计学基础的理解，并将为统计学家在处理他们在实践中遇到的问题时提供一个额外的工具。更直接的是，拟议的应用问题的成功解决将立即在应用领域取得成果，例如药品统计和计量。例如，美国食品和药物管理局(FDA)的指导文件详细说明了证明两种或两种以上药物配方等效性的分析程序。调查人员的目标是证明，基准方法将导致更有效的程序，这将导致成本和时间的节省，这是制药业的一个重要问题。在计量方面，国际度量局(BIPM)与国际标准化组织(ISO)联合出版了一份《测量不确定度表达指南》(GUM)，其中详细说明了美国NIST、英国NPL和德国PTB等国家计量机构应遵循的程序。计量学特有的一个问题是，每一次测量都会受到未知和不可知的系统误差的影响，这些误差往往比随机误差更大。量化这些不可知的系统误差的唯一方法是通过指定它们的主观分布。GUM规定了如何将计算中某些误差分量的标准偏差的基于数据的估计与其他误差分量的主观不确定度估计结合起来。研究人员的目的是证明，基准方法为实现这一目标提供了一种自然的方法。这样的结果可能会影响计量界修改和改进他们目前的程序。

In this proposal the investigators revisit Fisher's controversial fiducial argument with a modern set of questions in mind. This is motivated by the success of generalized inference as introduced by Tsui & Weerahandi (1989), which in fact leads to the same results as Fisher's fiducial inference (Hannig, Iyer & Patterson, 2006). The investigators do not attempt to derive a new ``paradox free theory of fiducial inference''. Instead, with minimal assumptions, the investigators present a new simple fiducial recipe that can be applied to conduct statistical inference via the construction of generalized fiducial distributions. This recipe is inspired by the concept of generalized pivotal quantity and is designed to be fairly easily applicable in many practical applications. It can be applied regardless of the dimension of the parameter space (i.e., including nonparametric problems), and it often leads to statistical procedures that are asymptotically exact and, more importantly, possess very good approximate small sample properties. The investigators propose to investigate theoretical properties of generalized fiducial distributions for statistical problems and apply their findings to various problems of broader interest. Systematic study of properties of generalized fiducial inference will increase our understanding of foundations of statistics and will give statisticians an additional tool to use when dealing with problems they encounter in practice. More directly, successful solution of the proposed applied problems will immediately bear fruit in the application areas, e.g., pharmaceutical statistics and metrology. For instance, the U.S. Food and Drug Administration (FDA) guidance document spells out analysis procedures for demonstration of equivalence of two or more drug formulations. The investigators aim to show that the fiducial approach will lead to more efficient procedures, which will result in cost and time savings, an important issue for the drug industry. In metrology, the International Bureau of Weights and Measures (BIPM) in conjunction with the International Organization for Standardization (ISO), has published a ``Guide to Expression of Uncertainty in Measurements" (GUM) which spells out the procedures to be followed by national metrological institutes such as NIST in the US, NPL in UK, and PTB in Germany. A problem that is unique to metrology is that every measurement is subject to unknown and unknowable systematic errors that are often larger than random errors. The only way to quantify these unknowable systematic errors is via specification of subjective distributions for them. The GUM specifies how to combine data-based estimates of standard deviations for some error components in the calculations and subjective estimates of uncertainty for other error components. The investigators aim to demonstrate that the fiducial method provides a natural approach for accomplishing this. Such results are likely to influence the metrology community in modifying and improving their current procedures.