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.
