High Dimensional Statistical Problems: Theory and Methods

这项研究将朝着在高维中存在混淆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进行。最初的工作包括基于两个新想法开发广义估计方程（GEE）方法的新工具：首先，研究者将开发一种最大信息方法来估计GEE中有害相关参数，其次，他将开发一种二次推理方法作为点估计器/标准误差方法的替代方法。发现第一种增强在协方差错配情况下提高了效率。第二种方法允许人们为估计方程框架开发类似ANOVA的模型选择技术，同时自动满足最大信息准则。然后，这些新工具将用于涉及高维相关数据的更广泛的具有挑战性的应用程序。研究的另一个主题是为最大似然失效的情况开发更广泛一致的非参数估计。现代统计学面临着日益复杂和精密的科学数据的爆炸式增长。此类数据的一个关键特征是高维数据。它的另一个特点是观测结果之间高度相互依赖。这项研究的目标是增强我们将这种相互依赖的影响与我们最感兴趣的数据特征分开的能力。这项研究的最初重点是处理纵向数据的一种流行方法，其中一个例子是对一组患者在一段时间内反复测量的反应变量。因此，如果我们希望了解响应变量如何受到解释变量的影响，那么患者内部观察相互依赖的结构就是我们必须适应的一个讨厌的特征。对于这种情况，将开发更有效和可靠的新方法，然后将它们扩展到具有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.