STATISTICAL INFERENCE WITH HIGH-DIMENSIONAL DATA

提出的研究将为高维数据的低维参数统计推断开发新的方法和算法。低维投影估计将在线性回归中得到进一步发展，并扩展到更一般的高维统计模型，包括广义线性模型、比例风险模型、大矩阵模型等。该项目将研究提出的估计量的一致性和渐近正态性，显著性检验，置信区间和区域，它们在最小费雪信息方面的效率，以及它们对多重调整的容忍度。本研究将直接连接半参数方法和高维数据领域，产生局部统一、高效的统计推断框架。由于信息技术的快速发展及其在现代科学实验中的应用，高维数据是当前统计研究和实践中非常感兴趣的一个领域。具有丰富高维数据的重要领域包括生物信息学、信号处理、神经成像、通信网络等。在许多这样的科学和工程应用中，未知的数量，以及问题的复杂性，是特征数量的函数：生物信息学中的遗传成分，神经成像中的大脑区域或体素，或者互联网中的计算机和路由器。在高维数据中，一个长期存在的挑战是在特征数量远远大于数据样本数量的情况下进行统计推断。现有的测试特征重要性的方法通常依赖于一个统一的信号强度假设：每个特征在对有效特征集的不确定性进行调整后，要么没有影响，要么比膨胀的噪声水平更强。然而，不幸的是，这种均匀信号强度的假设很少得到数据或基础科学的支持，特别是在生物学、医学、通信和社会网络中的应用。提出的研究将侧重于一种新的方法来解决上述长期存在的高维数据统计推断问题。它将开发实用的方法、高效的算法、统计软件和坚实的理论，用于测试低维特征函数的显著性和置信区域，即使在数据维数很高的情况下。在拟议的研究中开发的方法将与现代信息技术繁荣的共同应用直接相关。

The proposed research will develop new methodologies and algorithms for statistical inference of low-dimensional parameters with high-dimensional data. A low-dimensional projection estimator will be further developed in linear regression and extended to more general high-dimensional statistical models, including generalized linear models, the proportional hazards model, large matrix models and more. The project will investigate consistency and asymptotic normality of the proposed estimators, test of significance, confidence intervals and regions, their efficiency in terms minimum Fisher information, and their tolerance to multiplicity adjustments. This research will directly connect the fields of semi-parametric methods and high-dimensional data, producing a locally uniform and efficient framework of statistical inference. High-dimensional data is an area of intense current interest in statistical research and practice due to the rapid development of information technologies and their applications to modern scientific experiments. Important fields with an abundance of high-dimensional data include bioinformatics, signal processing, neural imaging, communications networks and more. In many such scientific and engineering applications, the number of unknowns, and thus the complexity of the problem, is a function of the number of features: genetic components in bioinformatics, brain regions or voxels in neural imaging, or computers and routers in the Internet. A longstanding challenge in high-dimensional data is statistical inference in situations where the number of features is far greater than the number of samples in the data. Existing methodologies for testing the significance of a feature commonly rely on a uniform signal strength assumption: Each feature has either no effect or an effect stronger than an inflated noise level after adjustments for the uncertainty of the set of effective features. However, this uniform signal strength assumption is, unfortunately, seldom supported by either the data or the underlying science, especially in applications in biology, medicine, and communication and social networks. The proposed research will focus on a new approach to the above mentioned longstanding problem of statistical inference with high-dimensional data. It will develop practical methods, efficient algorithms, statistical software, and solid theory for test of significance and confidence regions for low-dimensional functions of features, even when the dimension of data is high. The methodologies developed in the proposed research will be directly relevant to common applications where modern information technologies prosper.