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

该项目侧重于由医疗、工程和保险应用推动的三项主要统计活动。第一个是信号和图像的最优去噪和分解。 对人脑中的功能性磁共振图像（fMRI）的分析是一个具体的应用实例。不严格地说，fMRI是一个有噪声的时间序列图像，反映了测量过程中血液中的氧气水平。 因为这是关于处理血液中的氧气，所以时间序列包含对功能刺激的呼吸、心脏和神经反应。拟议的统计程序将允许医生和生物工程师分离和降噪这些组件，在开发诊断和治疗阿尔茨海默病和帕金森病的新方法中具有潜在的应用。第二个主题是噪声信号的大互协方差/相关矩阵的估计，元素的数量以百万计，样本大小仅为几百。这是用于研究细菌的芯片上芯片微阵列的统计分析中常见的问题。另一个应用是神经可塑性的研究，这是大脑根据新的经验和学习变化识别神经通路的能力。这将使医生能够创造新的方法来早期诊断和治疗中风，这是美国第四大死亡原因。第三个主题是从间接观测中自适应有效地估计风险率和生存函数，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新方法以及废水处理的创新技术和制定适应性寿命表的精算问题。(1)推进对非参数曲线估计的知识和理解，以开发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，并生成一系列更准确的估计量。初步结果表明，新的有效的估计，可以提出通过模仿神谕或通过不同的估计在频域中的聚合。最具挑战性和有益的结果是预期的多变量曲线，新的利率可以补救熟悉的灾难的多维性。预计将在微阵列的统计分析、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中应用。(2)根据小波方法和指数不等式，开发新的方法来推断噪声信号的大互协方差/相关矩阵，这些方法可应用于相关和非高斯观测。主要应用是研究人脑的神经可塑性。这使得医生和生物工程师能够根据新的经验和学习变化来观察神经通路的变化，并将其应用于治疗中风和其他脑部疾病。(3)改进非参数理论，为间接观测创建风险率和生存函数估计的有效方法，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.