Non-parametric Inferences and Related Topics

本研究涉及以下非参数问题：BUMP HUNTING问题，其中研究了单模态对多模态和相关估计量的检验；广义线性模型中平均响应函数的置信带投射追踪推理；以及RANDOM FIELDS，其中使用微分几何的大偏差和技术研究高斯随机场的最大值。这些问题除了方法论的发展和理论的分析外，还涉及到一些重要的计算。研究者还研究有效的算法，并将这些技术应用于空气污染数据和其他高维和分类数据。数据集的密度估计中的模态或起伏可能揭示数据的有趣结构。例如，在进化生物学中，它们可能表明某种物种将是稳定“自然选择”的主导因素，在高能物理学中，它们可能显示“部分波散射振幅”甚至新粒子的证据。本研究中研究的BUMP HUNTING技术有助于确定模式或凸起是数据下划线分布的真实特征还是仅仅是采样波动的结果。随机场和广义线性模型可以用来模拟空气污染数据和许多其他数据。投影追踪是一种强大的技术，它通过低维投影来搜索高维数据中有趣的结构。研究的另一个目标是确定基于这种技术的估计可能附加的置信度。

DMS 9626108 Sun This research involves the following non-parametric problems: BUMP HUNTING PROBLEMS, where tests of unimodality against multi-modality and related estimators are studied; CONFIDENCE BANDS for the mean response function in generalized linear models; inferences in PROJECTION PURSUIT; and RANDOM FIELDS, where the maxima of Gaussian random fields are studied using large deviations and techniques from differential geometry. All of these problems involve some nontrivial computations besides methodology development and theoretical analysis. The investigator also studies efficient algorithms and applies the techniques to air pollution data and other high dimensional and categorical data. Modes or bumps in a density estimate of a data set may reveal interesting structures of the data. For example, in evolutionary biology they may indicate that a certain kind of species will be a dominating factor in stabilizing "natural selection", and in high-energy physics they may show the evidence of "partial-wave scattering amplitudes" or even a new particle. BUMP HUNTING techniques studied in this research help decide whether the modes or bumps are real features of the underline distribution of the data or simply a result of sampling fluctuation. RANDOM FIELDS and GENERALIZED LINEAR MODELS can be used to model air pollution data and many other data. PROJECTION PURSUIT is a powerful technique that searches for interesting structures of a high dimensional data via lower dimensional projections. Another objective of the research is to determine the amount of confidence that may be attached to an estimate based on such techniques.