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。第一个问题是信号和图像的最优去噪和分解。分析人脑中的功能性磁共振图像(FMRI)就是一个特别的应用例子。简单地说，功能磁共振成像是一系列嘈杂的图像，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，时间序列包含呼吸、心脏和神经对功能刺激的反应。拟议的统计程序将使医生和生物工程师能够分离和去除这些成分，并可能应用于开发诊断和治疗阿尔茨海默氏症和帕金森氏病的新方法。第二个主题是对噪声信号的大互协方差/相关矩阵的估计，噪声信号的元素数量以数百万为单位，样本大小仅为数百。这是用于研究细菌的芯片芯片统计分析中常见的问题。另一个应用是研究神经可塑性，这是大脑根据新的经验和学习中的变化识别神经路径的能力。这将使医生能够创造新的方法来早期诊断和治疗中风，中风是美国第四大死亡原因。第三个主题是从间接观察中自适应和有效地估计危险率和生存函数，包括序贯对照试验和方案的新方法。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制定适应性生命表的精算问题。该项目集中于三个目标。(1)提高对非参数曲线估计的认识和理解，发展收缩局部极小极大估计的一般理论，使统计学家能够得到一个新的估计质量基准，并产生一系列更精确的估计量。初步结果表明，可以通过模仿先知或通过在频域中聚合不同的估计器来提出新的有效估计器。最具挑战性和回报的结果预计将出现在多变量曲线上，在那里，新的利率可以弥补人们熟悉的多维性诅咒。可应用于微阵列的统计分析、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。(2)基于小波方法和可应用于相依和非高斯观测的指数不等式，开发噪声信号的大互协方差/相关矩阵的新的推断方法。其主要应用是研究人脑的神经可塑性。这使医生和生物工程师能够基于新的经验和学习的变化观察神经路径的变化，并将其应用于中风和其他脑部疾病的治疗。(3)改进非参数理论，创建用于间接观察的风险比率和生存函数估计的有效方法，包括顺序对照实验和方案的新方法。

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.