Mathematical Sciences: Some Problems in Saddlepoint Approximations in Statistics

该项目的目标是开发新的鞍点理论和方法，扩展到迄今为止未被任何深度攻击的领域。特别要注意的是，在样本调查数据的分析中使用鞍点近似，在重采样问题中使用鞍点展开，在构造非参数函数估计的置信区域时使用一般二次形式的鞍点展开。对于许多采用复杂估计器的现代统计技术，在评估统计估计器的准确性和精密度方面存在重要的技术困难。例如，复杂的调查产生的数据通常需要复杂的数学形式来估计感兴趣的数量。对估计过程中固有的可变性量做出明确的陈述，受到这些估计量精确分布的难处的阻碍。接近的近似值可以取代这些分布；这项工作将利用鞍点理论来发展这种令人满意的近似分布。

The goal of this project is to develop new saddlepoint theory and methods, branching into areas not attacked in any depth to date. In particular, attention will be directed toward the use of saddlepoint approximations in the analysis of sample survey data, to saddlepoint expansions for resampling problems, and to saddlepoint expansions for a general quadratic form to be used in constructing confidence regions for nonparametric function estimation. For many modern statistical techniques employing sophisticated and often complicated estimators, there is an important technical difficulty in assessing the accuracy and precision of statistical estimators. For example, complex surveys yield data which usually require complicated mathematical forms for estimators of the quantities of interest. Making a clear statement about the amount of variability inherent in the estimation process is hindered by the intractability of the exact distributions of these estimators. Close approximations may replace these distributions; this work will utilize saddlepoint theory to develop such satisfactory approximate distributions.