High Dimensional Statistical Problems: Theory and Methods

DMS-9870193Lindsay这项研究将针对这样一个广泛的主题：在存在高维混杂变异的情况下，恢复有关兴趣变异的统计信息。最初的工作包括开发基于两个新想法的广义估计方程(GEE)方法的新工具：第一，研究人员将开发一种估计广义估计方程中滋扰相关参数的最大信息方法，第二，他将开发一种二次推理方法，作为点估计/标准误差方法的替代。第一个增强被发现在协方差错误指定的情况下提高了效率。第二种方法允许开发类似于ANOVA的模型选择技术，用于估计方程框架，同时自动满足最大信息标准。然后，这些新工具将用于更广泛的具有挑战性的应用程序，涉及高维相关数据。这项研究的另一个主题是为最大似然失败的情况开发更广泛一致的非参数估计。现代统计学面临着越来越复杂和复杂的科学数据的爆炸性增长。这类数据的关键特征之一是它是高维的。它还具有观测之间高度相互依存的特点。这项研究的目标是提高我们将这种相互依赖的影响与我们最感兴趣的数据的特征区分开来的能力。这项研究最初的重点是一种处理纵向数据的流行方法，其中一个例子是在一组患者身上重复测量的反应变量。因此，如果我们想要了解解释变量如何影响反应变量，患者内部观察的相互依赖结构就是一个令人讨厌的特征。新的方法将被开发出来，在这种情况下更有效和可靠，然后它们将被推广到其他相关结构的问题。

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.