Two Problems in Statistical Inference

当使用可信区间(CI)或假设检验(HT)对参数进行推断时，CI通常提供关于参数的更多信息，但很难构造；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准确信息。首席调查者(PI)试图解决这两个问题(至少在一定程度上)：i)提供基于覆盖概率的CI的构造方法；ii)HT的推广。自从1814年拉普拉斯首次提出CI以来，人们已经做出了许多努力来推导CI。构造CI的方法有五种：枢轴量法、检验倒排法、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，而覆盖概率是CI定义中所需要的全部。开发这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造包括最小CI(任何其他CI的子集)在内的最优CI，并且最小区间自动最小化期望长度和错误覆盖概率。在不同的场景下，PI将使用PI在2006年引入的集合包含准则来构造最小或允许的CI。传统的HT只处理二选一问题。然而，大多数应用程序都涉及多项选择问题。在提案的第二部分，PI将推广HT程序，使人们能够在两个以上相互排斥的索赔中做出选择。这可以通过以下方式来完成：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确定哪个索赔被测试为备选方案，最后对所选索赔进行传统测试。这一新程序为解决任何多项选择问题提供了灵活性。将讨论各种应用，包括传统的问题，如方差分析、模型选择、检测质量控制中的微小漂移，以及一些开放的问题，包括非正交饱和设计中的有效效应检测。简而言之，几乎所有的测试问题，除了那些片面的替代方案，都可以使用新的程序重新考虑，并期望得到不同的、更有效的结果。参数是描述感兴趣总体的整个分布的某个量，关于该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服用某种药物后出现改善的所有患者(人群)的比例(参数)。作为一个参数的两种主要的统计推断工具，可信区间(CI)处理“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牢固。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，这一问题的良好解决方案将非常有助于建立一种新开发的药物比对照药物更安全和更有效的优势。作为另一种情况，大剂量的药物通常会有严重的副作用。因此，确定有效药物的最小剂量水平对患者来说是一个重要的问题。这涉及到将不同剂量水平的几种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/或很高的准确率做出估计、预测和决策。正在进行的研究试图从根本上提高对统计学的理解，并将导致上述两个问题作为直接应用的更好或最优解决方案。更具体地说，将基于覆盖概率构建较短的可信区间，并且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.