Mathematical Sciences: Investigations in Mathematical Statistics

统计分析是科学技术研究和发展的重要组成部分。这些分析具有广泛的数学联系，尽管许多问题也需要适应特定应用的特定技术。本研究计划涉及对这一常见数学结构各方面的调查。非参数函数估计是非参数函数估计的研究热点之一，它在图像处理和电子信号处理与分析中有着广泛的应用。目前的研究将为这类程序确立理想的一般性质。然后，本文将研究通常使用的程序在多大程度上具有这些属性，如果没有这些属性，则应如何修改它们。研究的另一个目标是建立一个通用理论，解释如何构建给定置信度的最小可能置信度集。还将调查对具体案件的影响。第三个重点是评估概率预测的一般方法，例如预测降雨的机会。对于单个预测者预测二项（=“是”或“否”）事件的情况，已经发展了相当多的理论。对于多项事件和由群体一致意见而不是由一个人作出的预测，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.