Nonparametric Curve Estimation in the Presence of Nuisance Functions

本文的研究重点是在存在干扰函数的情况下，发展自适应非参数曲线估计的统计方法、理论和方法。它的动机是生物和医学应用。研究者主要研究四类被考虑的问题。每一个都是根据一个估计器如何匹配一个知道潜在的麻烦函数的oracle的性能来分类的。它们是：(a) Estimator可以匹配oracle。一个例子是一个回归问题，平滑设计密度是干扰函数。(b)如果有补充观测，估计器可以匹配oracle。例如，在需要补充观测来估计测量误差分布的预测器中，有测量误差的反卷积和回归。(c)估计器无法匹配oracle。一个例子是当干扰回归函数不够光滑时回归误差密度的估计。(d)上述环境的混合。建议为上述统计问题的特定应用开发一种自适应估计的一般理论：缺失，分层和删除数据，隐藏成分，涉及连续和名义或有序分类变量的混合多元模型，以及时间序列。理论结果被测试并应用于芯片芯片微阵列和超快速功能磁共振成像的分析。研究的主要焦点是创建自适应的统计程序，可以在妨害函数的存在下工作。本研究的动机和测试是在统计分析中得到充分理解的应用：(i)用于在细菌基因组中寻找调节蛋白结合位点的ChIP-on-chip芯片微阵列。蛋白质和DNA之间的相互作用是生命的基础。它们促进和介导基因表达、DNA复制和修复。提出的ChIP-on-chip芯片微阵列的统计分析指出了蛋白质- dna结合位点的确切位置。由于统计分析不需要测量干扰函数，这使得微阵列实验更便宜，更快，更准确。（ii）超快速功能性磁共振成像，有助于了解衰老和脑部疾病，如阿尔茨海默病和帕金森病。超高速功能磁共振成像是一项令人兴奋的研究大脑功能的新技术，其时间分辨率为50毫秒。这一分辨率揭示了大脑中的神经元和生理活动。提出的统计分析方法对滋扰函数具有鲁棒性，可以去噪情绪动力反应，研究记忆、语言和情感等认知功能，并绘制人类大脑的生理活动图。

The proposal focuses on developing statistical methodology, theory and methods of adaptive nonparametric curve estimation in the presence of nuisance functions. It is motivated by biological and medical applications. The investigator studies four main classes of considered problems. Each is classified by how an estimator can match the performance of an oracle knowing underlying nuisance functions. They are: (a) Estimator can match oracle. An example is a regression problem with a smooth design density being nuisance function. (b) Estimator can match oracle if complementary observations are available. Examples are deconvolution and regression with measurement errors in predictors where complementary observations are needed to estimate distribution of measurement error. (c) Estimator cannot match oracle. An example is estimation of the density of regression error when a nuisance regression function is not sufficiently smooth. (d) A mixture of the above-formulated settings. It is proposed to develop a general theory of adaptive estimation for the aforementioned classes of statistical problems with particular applications to: missing, stratified and censored data, hidden components, mixed multivariate models involving continuous and nominal or ordinal categorical variables, and time series. Theoretical results are tested and applied to the analysis of ChIP-on-chip microarrays and ultra-fast fMRI.The primary focus of the research is to create adaptive statistical procedures which can work in the presence of nuisance functions. This research is motivated by and tested on well-understood applications in the statistical analysis of: (i) ChIP-on-chip microarrays used to find regulatory protein binding sites in a bacterial genome. Interactions between protein and DNA are fundamental to life. They facilitate and mediate gene expression, DNA replication and repair. The proposed statistical analysis of ChIP-on-chip microarrays points on exact location of protein-DNA binding sites. Because the statistical analysis does not require measuring of nuisance functions, it makes microarray experiments cheaper, faster and more accurate. (ii) Ultra-fast functional magnet resonance images, which help in understanding aging and brain diseases such as Alzheimer's and Parkinson's Diseases. Ultra-fast fMRI is an exciting new technology for studying brain functions with the temporal resolution of 50 milliseconds. This resolution sheds light on both neurons and physiological activities in the brain. Proposed statistical analysis, which is robust to nuisance functions, can denoise emodynamic responses, study cognitive functions like memory, speech and emotion, and create a map of physiological activities of the human brain.