Data-Analytic Modeling for High-Dimensional Data

这项研究涉及各种新的数据分析技术，用于处理和建模来自许多科学学科的高维数据。这些模型的范围从流行病学和环境统计学的变系数模型、金融的时间非均匀扩散模型、眼科、营销和生物统计学的曲线和图像数据，到统计和工程中的小波应用。这项研究极大地提高了分析这些复杂高维数据的工具和软件的可用性。这项研究涵盖了广泛的方法学发展和基础研究。特别是，集中讨论了四个相互关联的主题。首先，提出了一种变系数模型下有效估计的两步策略。当系数函数允许不同程度的光滑度时，这特别有用。所提出的方法将通过精心设计的渐近理论和仿真来证明。所提出的方法扩展到纵向研究和基于似然的模型，如广义线性模型。其次，提出了时间非均匀扩散模型来模拟期限结构动态和资产定价。这些非参数模型旨在解决过于简化的参数模型的不足。各种数据分析建模技术被提出，并被理论研究和实证研究所证实。第三，讨论了小波在各种统计问题中的应用。所提出的方法源于统计建模的前瞻性，并基于惩罚似然思想。我们的目的是扩大小波在统计学中的适用性，并提高一些现有技术的效率。特别是，提出了基于似然模型、稳健性、相关数据去噪和加性建模的研究方法。最后，对曲线和图像数据处理中的一些统计推断问题进行了研究。

9803200Jianqing FanThis research involves a variety of new data-analytic techniques for processing and modeling high dimensional data that arise from many scientific disciplines. These range from varying-coefficient models from epidemiology and environmental statistics, time-inhomogeneous diffusion models from finance, curves and images data from ophthalmology, marketing and biostatistics, to wavelet applications in statistics and engineering. The research enhances significantly the availability of tools and software for analyzing these complicated high-dimensional data.The research covers a wide array of methodological developments and foundational research. In particular, four inter-related topics are focussed. Firstly, a two-step strategy is proposed for efficient estimation in varying-coefficient models. This is particularly useful when coefficient functions admit different degrees of smoothness. The proposed methods will be justified by carefully formulated asymptotic theory and simulations. The proposed methods extends to longitudinal studies and likelihood-based models such as generalized linear models. Secondly, time-inhomogeneous diffusion models are proposed to model term-structure dynamics and asset pricing. These nonparametric models aim at addressing inadequacy of over simplified parametric models. Various data-analytic modeling techniques are proposed and justified by both theoretical studies and empirical research. Thirdly, wavelet applications to a variety of statistical problems are discussed. The proposed approaches derive from statistical modeling prospective and are based on penalized likelihood idea. Our aim is to expand the applicability of wavelets in statistics and to improve efficiency of some existing techniques. In particular, likelihood based models, robustness, denoising correlated data and additive modeling are proposed for investigation. Finally, some statistical inference problems arising from processing curves and images data are investigated.