基于充分降维方法的高维数据假设检验问题的研究

The scale and complexity of data sets have increased drastically in light of the development in modern technology. High-dimensional data that involve a large amount of variables are nowadays routinely generated and collected in areas such as environmental studies, human health and medical research, financial markets and second-generation internet. High-dimensional data pose many challenges for statisticians and provide considerable momentum in the statistics community to develop new theories and methodologies. Sufficient dimension reduction methodology effectively transforms a high dimensional data problem to a low dimensional one, and thus facilitates many existing statistical methods which used to be hindered by the curse of dimensionality. This project proposes a new paradigm that synthesizes and broadens the theories and methodologies of hypotheses testing for high dimensional data with sufficient dimension reduction methods. We focus mainly on the following three topics: 1) Marginal dimension tests with second order parametric sufficient dimension reduction methods; 2) Testing predcitors contributions based on sufficient dimension reduction methods for ultrahigh dimensional semiparametric models; 3) Deciding the strucutre dimension and feature selection for high dimensional function data based on hypotheses testing. The results from this project should be useful for high dimensional problems involved in areas such as econometrics and bioinformatics.

随着现代科学技术的发展，数据越来越凸显高维与复杂态势。高维数据频繁见诸于环境科学、医学研究、金融市场以及第二代互联网当中。高维数据给统计学家提出巨大挑战的同时，也给予了统计学界进一步深化发展统计理论与方法的契机。充分降维通过有效将高维变量转化为其低维线性组合，避免了"维数祸根"的困扰，使得进一步统计分析成为可能。本项目致力于通过充分降维方法系统构建并深化高维数据模型下的假设检验理论。我们将着重展开如下三个方面的研究：1）基于二阶参数化降维方法的结构维数检验理论和方法；2）超高维半参数模型下变量显著检验理论和方法；3）高维函数型数据模型的结构维数检验以及特征显著性检验。本项目的研究成果必将为分析和处理金融经济学以及生物信息学等领域所面临的高维数据提供有力的方法保证和理论支持。

当代统计分析常常面临来自各领域的高维数据，有些高维数据的维数甚至远远大于样本量。为了更有效的挖掘高维数据的内在信息，本课题着力从假设检验的角度并结合充分降维方法展开研究，并将研究的内容进一步拓展至相关高维数据分析领域。我们主要研究了条件独立性检验、变量边际显著性检验、多总体指标模型指标检验三方面重要的假设检验问题。基于我们构建的假设检验理论和方法，我们还进一步研究了高维数据领域的变量筛选和选择问题。除此之外，我们还研究了高维数据分析领域里的其它重要相关议题，比如惩罚充分降维方法、稳健型降维方法以及基于充分降维方法的统计建模问题。在项目资助下，我们在顶级统计学期刊Biometrika, Journal of the American Statistical Association等期刊上发表了数篇高质量学术论文，并多次参与国内外相关学术会议汇报项目研究成果。

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