Mathematical Sciences: Investigations in Mathematical Statistics

统计分析是科学技术研究与开发的重要组成部分。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。这项研究计划涉及对这一共同数学结构的各个方面的调查。研究的重点之一是非参数函数估计，它包括在图像处理和电子信号处理和分析中的各种应用。目前的研究将为这类程序建立理想的一般性质。然后，它将审查常用程序在多大程度上具有这些特性，以及在没有这些特性的情况下，应该如何修改它们。这项研究的另一个目标是一个解释如何构造给定置信度的最小可能置信集的一般理论。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，例如预测的降雨机会。对于单个预报员预测二项(=“是”或“否”)事件的情况，已经发展了相当多的理论。将为多项活动的情况和根据集体共识而不是由一个人产生的预测提出延期建议。

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.