Topics in Nonparametric Statistics: Faster Minimax Rates, Large-p-Small-n Cross-Correlation Matrices, Survival Analysis

该项目侧重于医疗、工程和保险应用所推动的三项主要统计活动。首先是信号和图像的最优去噪和分解。分析人类大脑中的功能磁共振图像（fMRI）是应用的一个特殊例子。粗略地说，功能磁共振成像是一个嘈杂的时间序列图像，它反映了测量期间血液中的氧气水平。因为这是关于处理血液中的氧气，时间序列包含呼吸、心脏和神经对功能性刺激的反应。拟议的统计程序将允许医生和生物工程师对这些成分进行分离和去噪，在开发诊断和治疗阿尔茨海默病和帕金森病的新方法方面具有潜在的应用。第二个主题是对噪声信号的大型交叉协方差/相关矩阵的估计，其中元素的数量为数百万，样本量仅为数百。这是用于研究细菌的芯片微阵列的统计分析中常见的问题。另一个应用是对神经可塑性的研究，即大脑根据新的经验和学习中的变化识别神经通路的能力。这将使医生能够创造出早期诊断和治疗中风的新方法，中风是美国第四大死亡原因。第三个主题是间接观察的自适应和有效的危险率和生存函数估计，包括顺序控制实验和协议的新方法。这项研究的动机是肺癌和乳腺癌的放射和药物治疗的新方法，以及废水处理的创新技术和开发适应性生命表的精算问题。该项目侧重于三个目标。(1)提高对非参数曲线估计的认识和理解，发展一种缩小局部极大极小估计的一般理论，使统计学家能够获得估计质量的新基准，并产生一系列更准确的估计器。初步结果表明，可以通过模拟预言器或在频域上聚合不同的估计器来提出新的高效估计器。最具挑战性和最值得期待的结果是多元曲线，其中新的速率可以弥补熟悉的多维诅咒。预计将应用于微阵列统计分析、功能磁共振成像、癌细胞的放射和药物治疗以及废水处理的创新技术。(2)基于可应用于相关和非高斯观测的小波方法和指数不等式，为噪声信号的大交叉协方差/相关矩阵开发新的推理方法。主要应用于研究人脑的神经可塑性。这使得医生和生物工程师能够观察到基于新经验的神经通路的变化，以及应用于中风和其他脑部疾病的学习变化。(3)改进非参数理论，建立有效的间接观测风险率和生存函数估计方法，包括顺序控制实验和协议的新方法。

The project focuses on three main statistical activities motivated by medical, engineering and insurance applications. The first one is the optimal denoising and decomposition of signals and images. Analysis of functional magnet resonance images (fMRI) in the human brain is a particular example of application. Loosely speaking, fMRI is a noisy time series of images which reflects the level of oxygen in blood during measurements. Because this is about dealing with oxygen in blood, the time series contains respiratory, cardiac and neural response to functional stimuli. The proposed statistical procedure will allow doctors and bioengineers to separate and denoise these components, with potential applications in developing new methods for diagnosis and treatment of Alzheimer's and Parkinson's Diseases. The second topic is the estimation of large cross-covariance/correlation matrices for noisy signals, with the number of elements in millions and sample sizes of just several hundreds. This is a familiar problem in statistical analysis of Chip-on-chip microarrays used to study bacteria. Another application is the study of neural plasticity, which is the ability of the brain to recognize neural pathways based on new experience and change in learning. This will allow physicians to create new methods for early diagnosis and treatment of stroke which is the 4th leading cause of death in the US. The third topic is adaptive and efficient estimation of hazard rate and survival function from indirect observations, including new methodology of sequentially controlled experiments and protocols. This research is motivated by new methods of radiation and drug therapy for lung and breast cancers as well as by innovative technologies of waste-water treatment and the actuarial problem of developing adaptive life tables.The project focuses on three objectives. (1) Advance knowledge and understanding of nonparametric curve estimation to develop a general theory of shrinking local minimax estimation that allows statisticians to get a new benchmark for the quality of estimation and generate a family of more accurate estimators. Preliminary results indicate that new efficient estimators can be proposed either via mimicking oracles or via aggregation of different estimators in frequency domain. Most challenging and rewarding results are expected for multivariate curves where new rates can remedy the familiar curse of multidimensionality. Applications in statistical analysis of microarrays, fMRI, radiation and drug therapy of cancer cells, and innovative technologies of waste-water treatment are expected.(2) Develop new methods of inference for large cross-covariance/correlation matrices for noisy signals, based on wavelet methods and exponential inequalities that can be applied to dependent and non-Gaussian observations. The main application is the study of neural plasticity of the human brain. This allows doctors and bioengineers to observe changes in neural pathways based on new experience and change in learning with applications to treatment of stroke and other brain diseases.(3) Improve nonparametric theory and create efficient methods of hazard rate and survival function estimation for indirect observations, including a new methodology of sequentially controlled experiments and protocols.