Mathematical Sciences: Custom-Designed Confidence Set Construction

该项目的目标是通过使用改进的和定制设计的置信集来提高实验人员可用的统计推断方法的质量。虽然很明显，对实验人员来说，一组参数的置信界限比问题的假设检验公式的答案更能提供信息，但在许多本质上基本的统计推断问题中，当前构建置信集的技术要么是粗糙的，要么是低效的，要么是不存在的。这可能是因为一个复杂的分布设置，因为真正感兴趣的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，如方向决定。在许多情况下，实验者有一些明确定义的目标，可以正确和明智地用来确定一个敏感和强大的推理方法，但会发现置信集目前只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可能的更充分、更完整的推理。重要的是发展这些技术并使其广泛使用，因为目前正在作出许多决策类型推断，例如在比较药物试验中，正在报告决策结果（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和药物B疗效差异的置信区间）也可以获得，但没有加以利用。本研究将考虑改进当前数据分析技术的方法。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断科学开发处理这种不确定性的改进方法。该调查将考虑实验设计或数据收集方案，以及数据分析。这项研究的主要目的将是发展新的统计分析结果的呈现方式，这些统计分析的目的是使外行人能够清楚和直观地理解。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、医学研究和其他重要领域的研究人员和实验人员有用。

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.