Two Problems in Statistical Inference

当使用置信区间（CI）或假设检验（HT）进行关于参数的推断时，通常CI提供关于参数的更多信息，但难以构造;而HT具有相对容易的构造，但不像CI那样提供关于参数位置的精确信息。主要研究者（PI）试图解决这两个问题（至少在一定程度上）通过i）提供一种基于覆盖概率的CI构造方法，以及ii）HT的推广。自从拉普拉斯在1814年首次提出CI以来，人们一直在努力推导CI。CI的构建方法主要有五种：关键量法、检验倒置法、保证区间法、贝叶斯法和不变性。但这些都不是基于覆盖概率的分析，而这正是CI定义所需要的。制定这样一种方法是该提案的目标之一。事实上，通过关注覆盖概率，可以在某类区间内构建最佳CI，包括最小CI（任何其他CI的子集），并且最小区间自动最小化预期长度和错误覆盖概率。PI将使用PI在2006年引入的集合包含标准，在不同的场景下构建最小或可接受的CI。传统的HT只处理二选一问题。然而，大多数应用程序涉及多项选择问题。在提案的第二部分，PI将推广HT程序，以便能够在两个以上的互斥索赔中做出选择。这可以通过首先将基本备选方案划分为多个索赔并相应地划分样本空间，然后使用观察到的数据来决定哪个索赔作为备选方案进行测试，最后对选定的索赔进行传统测试来完成。这个新的程序提供了灵活性，以解决任何多项选择问题。各种应用程序将得到解决，包括传统的问题，如方差分析，模型选择，检测质量控制中的小变化，和一些开放的问题，包括检测非正交饱和设计中的积极影响。简而言之，几乎所有的测试问题，除了那些有一个片面的选择，可以重新考虑使用新的程序，不同的，更有效的结果是预期的。参数是描述感兴趣的总体的整个分布的一定量，并且关于参数的推断是统计学中的基本问题之一。一个简单但非常有用的例子是估计所有患者（人群）在服用某种药物后表现出改善的比例（参数）。作为参数的两个主要统计推断工具，置信区间（CI）解决“什么”类型的问题，假设检验（HT）回答“是”或“否”类型的问题。尽管近年来统计理论和应用取得了巨大的进步，但统计学的基础并不像它应该的那样坚实。一些基本问题，包括两个比例的比较，仍然没有一个理想的解决方案。 然而，这个问题的一个很好的解决方案将非常有助于建立一个新开发的药物比控制更安全，更有效的优越性。在另一种情况下，高剂量的药物通常具有严重的副作用。因此，确定有效药物的最低剂量水平对患者来说是一个重要问题。这涉及将不同剂量水平的几个比例与对照组的常见比例进行比较。统计学的主要任务是根据观察到的数据，以测量的精度和/或高概率进行估计，预测和决策。正在进行的研究是试图从根本上提高对统计的理解，并将导致更好或最佳的解决方案，上述两个问题作为直接应用。 更具体地说，短的置信区间将构建覆盖概率的基础上，新提出的测试程序将能够处理多项选择问题。

When making inferences about parameters using the confidence interval (CI) or the hypothesis test (HT), typically, the CI provides more information about the parameter, but is hard to construct; while the HT has a relatively easy construction, but does not provide precise information about the location of parameter as the CI does. The principle investigator (PI) tries to resolve these two problems (at least to a certain degree) by i) providing a construction method of the CI based on coverage probability, and ii) a generalization for the HT. There have been many efforts to derive a CI since it was first proposed by Laplace in 1814. There are five methods for the CI construction: pivotal quantities, inversion of tests, guarantee intervals, Bayesian method and invariance. But none of these is based on the analysis of coverage probability, which, however, is all one needs in the definition of CI. The development of such a method is one goal of the proposal. In fact, by focusing on the coverage probability, optimal CIs, including the smallest CI (a subset of any other CI), can be constructed within a certain class of intervals, and the smallest interval automatically minimizes the expected length and the false coverage probability. The PI will construct the smallest or admissible CI's using the criterion of set inclusion introduced by PI in 2006 under different scenarios. The traditional HT only deals with a two-choice problem. However, most applications involve a multiple-choice problem. In the second part of the proposal, the PI will generalize the HT procedure so that one is able to make a choice among more than two mutually exclusive claims. This can be done by first partitioning the basic alternative into multiple claims and partitioning the sample space correspondingly, then using the observed data to decide which claim is tested as the alternative, and finally conducting a traditional test for the selected claim. This new procedure provides flexibility to solve any multiple-choice problem. Various applications will be addressed, including traditional problems, such as analysis of variance, model selection, detecting small shifts in quality control, and some open problems, including the detection of active effects in nonorthogonal saturated designs. In short, almost all testing problems, except for those with a one-sided alternative, can be reconsidered using the new procedure, and different, more efficient results are expected. A parameter is a certain quantity that describes the entire distribution of a population of interest, and inference about the parameter is one of the fundamental problems in Statistics. A simple but very useful example is to estimate the proportion (the parameter) of all patients (the population) who show improvement after taking a certain drug. As two major statistical inference tools for a parameter, the confidence interval (CI) addresses the "what" type of question and the hypothesis test (HT) answers the "yes" or "no'' type of question. In spite of the tremendous progress in statistical theory and applications in recent years, the foundation of Statistics is not as solid as it should be. Some basic problems, including the comparison of two proportions, still do not have an ideal solution. However, a fine solution for this problem would be very helpful to establish the superiority of a newly developed drug over the control more securely and more efficiently. As another case, a high dose of a drug typically has a severe side effect. So identifying the minimum dose level of a drug that is effective is an important issue for patients. This involves the comparison of several proportions for different dose levels with a common proportion of the control group. The main task of Statistics is to make estimations, predictions and decisions with measured precision and/or high probability of being correct based on the observed data. The ongoing research is an attempt to improve the understanding of Statistics from the root, and will lead to better or optimal solutions for the two problems mentioned above as direct applications. More specifically, short confidence intervals will be constructed based on coverage probability, and the newly proposed testing procedure will be able to handle the multiple-choice problem.