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.
