Mathematical Sciences: Custom-Designed Confidence Set Construction

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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.