Mathematical Sciences: Investigations in Mathematical Statistics

统计分析是科学和技术研究与发展的一个重要组成部分。这些分析有着广泛的数学联系，尽管许多问题也需要适应特定应用的特定技术。这项研究计划涉及到这个共同的数学结构方面的调查。研究的一个焦点是非参数函数估计，它包括在图像处理和电子信号处理和分析中的各种应用。 目前的研究将建立理想的一般性质，这样的程序。然后，它将研究在何种程度上，在共同使用的程序有这些属性，如果他们不，他们应该如何修改。 研究的另一个目标是一个通用的理论，解释如何构建最小可能的置信度给定的信心集。还将调查对具体案件的影响。第三个重点是评估概率预报的一般方法，例如预测下雨的机会。 对于单一预测者预测二项式（=“是”或“否”）事件的情况，已经发展了相当多的理论。 对于多项事件的情况和由群体共识而不是由一个人产生的预测，将提出扩展。

Statistical analysis is an essential component of scientific and technical research and development. These analyses have broad mathematical links in common, although many problems also require specific techniques adapted to the particular application. This research proposal involves investigations into aspects of this common mathematical structure. One focus of the research is nonparametric function estimation, which encompasses varied applications in image processing and in electronic signal processing and analysis. The current research will establish desirable general properties for such procedures. It will then examine to what extent the procedures in common use have these properties and, where they do not, how they should be modified. Another goal of the research is a general theory which explains how to construct the smallest possible confidence sets of given confidence. The implications for specific cases will also be investigated. A third focus is on general methods for evaluating probabilistic forecasts, such as the predicted chance of rain. Considerable theory has been developed for the case of a single forecaster predicting a binomial (="yes" or "no") event. Extensions will be proposed for the case of multinomial events and for forecasts produced by group consensus rather than by one individual.