Mathematical Sciences: Custom-Designed Confidence Set Construction

该项目的目标是通过使用改进的和定制的置信集，提高实验者可用的统计推断方法的质量。虽然很明显，一组参数的置信度对实验者来说比对问题的假设检验公式的答案更具信息性，但在许多基本的统计推断问题中，当前的置信度集构造技术要么粗糙、效率低下，要么根本不存在。这可能是因为复杂的分布设置，因为感兴趣的真实参数是其他基本参数的函数，或者因为实验者选择强调某些类型的信息，例如方向决策。在许多情况下，实验者有一些明确定义的目标，这些目标可以正确和明智地用于确定敏感和强大的推理方法，但会发现置信集目前仅适用于标准的不太具体的推理方法。与这些特定推理程序相关联的定制设计置信集的构建将使实验者能够做出比目前通常可能的更全面和更完整的推理。重要的是开发这些技术并使它们广泛可用，因为目前正在进行许多决策类型推断，例如在比较药物试验中，其中报告决策结果(例如，药物A比药物B更有效)，并且实际上，其他免费信息(如药物A和B的疗效差异的可信区间)也是可用的，但没有得到利用。这项研究将考虑如何改进当前的数据分析技术。对数据的测量和随后对这些数据的分析不可避免地涉及一些不确定性，本研究的目的是通过统计推理科学开发处理这种不确定性的改进方法。这项调查将考虑实验设计或数据收集方案，以及数据分析。这项研究的主旨将是开发新的方法来显示统计分析的结果，这些方法旨在使非专业人员能够清楚和初步地理解。此外，这些方法必须是准确的，它们必须有效地总结所有可用的信息。这些新技术将对工程、质量控制、临床试验、医学研究等重要领域的研究人员和实验者有用。

The objective of this project is to improve the quality of statistical inference methods available to an experimenter through the use of improved and custom-designed confidence sets. While it is clear that confidence bounds on a set of parameters are more informative to an experimenter than the answer to a hypothesis test formulation of the problem, there are many essentially basic statistical inference problems where current techniques for confidence set construction are either crude, inefficient or nonexistent. This may be because of a complicated distributional set-up, because the real parameters of interest are functions of other basic parameters, or because the experimenter has chosen to emphasize certain kinds of information such as directional decisions. In many settings, the experimenter has some clearly defined objectives which can rightly and sensibly be used to determine a sensitive and powerful inference method, yet will find out that confidence sets are currently only available for standard less specific inference approaches. The construction of custom-designed confidence sets associated with these specific inference procedures will enable the experimenter to make a fuller and more complete inference than is often currently possible. It is important to develop these techniques and to make them widely available since many decisions type inference are currently being made, such as in comparative drug trials, in which the decision results are being reported (e.g., drug A is more effective than drug B) and where, in fact, additional free information (such as a confidence interval for the difference in the efficacies of drugs A and B) is also available but is not being utilized. This research will consider ways to improve current data analysis techniques. The measurement of data and the subsequent analysis of such data inevitably involves some uncertainty, and the objective of this research is to develop improved ways of dealing with this uncertainty through the science of statistical inference. This investigation will consider both the design of experiments or data collection schemes, together with the analysis of the data. The main thrust of the research will be to develop new ways of presenting the results of statistical analyses which are designed to be clear and intiutively comprehensible to the lay person. Moreover, these methods must be accurate and they must efficiently summarize all the available information. These new techniques will be useful for researchers and experimenters in areas such as engineering, quality control, clinical trials, medical studies and other important fields.