High Dimensional Statistical Problems: Theory and Methods

本研究将针对在高维中存在混杂变异的情况下恢复有关感兴趣变异的统计信息这一广泛主题。 初步的工作涉及开发新的工具的方法，广义估计方程（GEE）的基础上两个新的想法：首先，调查员将开发一个最大的信息方法来估计滋扰相关参数GEE，第二，他将开发一个二次推理方法作为替代点估计/标准误差的方法。 第一个增强已被发现，以增加协方差误指定下的效率。 第二个允许一个开发的ANOVA模型选择技术的估计方程框架，同时自动满足最大信息标准。 这些新工具将被用于更广泛的涉及高维相关数据的具有挑战性的应用。 研究的另一个主题是发展更广泛一致的非参数估计的情况下，最大似然fuls.Modern统计面临着日益复杂和精密的科学数据爆炸。 这种数据的一个关键特征是它是高维的。 它的特点还在于观察之间具有高度的相互依赖性。 研究的目标是提高我们将这种相互依赖性的影响与我们最感兴趣的数据特征分开的能力。 本研究的最初重点是处理纵向数据的流行方法，其中一个例子是随时间推移对一组患者重复测量的响应变量。 因此，如果我们想了解解释变量如何影响响应变量，那么病人体内观察结果的相互依赖性结构就是一个令人讨厌的特征。 新的方法将被开发，更有效和可靠的设置，然后他们将被扩展到相关结构的其他问题。

DMS-9870193LindsayThe research will be directed toward the broad theme of recovering statistical information about variation of interest in the presence of confounding variation in high dimensions. The initial work involves developing new tools for the methodology of Generalized Estimating Equations (GEE) based on two new ideas: first, the investigator will develop a maximum information approach to the estimation of nuisance correlation parameters in GEE, and second, he will develop a quadratic inference methodology as an alternative to the point estimator/standard error approach. The first enhancement has been found to increase efficiency under covariance misspecification. The second allows one to develop ANOVA like model selection techniques for the estimating equation framework while automatically meeting the maximum information criterion. These new tools will then be used in a wider range of challenging applications involving high dimensional correlated data. An additional theme of the research is the development of more widely consistent nonparametric estimators for situations in which maximum likelihood fails.Modern statistics is faced with an explosion of increasing complex and sophisticated scientific data. One of the key features of such data is that it is high-dimensional. It is also characterized by having high degrees of interdependence between observations. The goal of the research is to enhance our ability to separate the effects of this interdependence from the features of the data that we are most interested in. The initial focus of this research is on a popular method of dealing with longitudinal data, an example of which would be response variables that are measured repeatedly over time on a group of patients. The structure of the interdependence of observations within a patient is then a nuisance feature which we must adapt to if we wish to learn how the response variables are affected by explanatory variables. New methods will be developed that are more efficient and reliable for this setting, and they will then be extended to other problems with correlated structures.