CAREER: Asymptotic Statistical Decision Theory and Its Applications

渐近等价理论是建立各种统计模型之间联系的理论，是吕西安·勒卡姆最重要的统计学贡献之一。如果两个模型是渐近等价的，所有的渐近最优统计估计量可以从一个模型转移到另一个模型。建立渐近等价的一个基本原则是用一个更容易处理的统计模型来逼近一个复杂的统计模型。高斯定位模型是一种易于处理的模型，它捕获了许多统计设置的本质。研究者研究了使用改进的分位数耦合不等式和新的方差稳定变换将一般非参数估计转换为高斯回归的明确和实用程序。其他统计问题可以通过与泊松过程模型相联系来更好地理解。研究了密度估计的无限可分近似及其与非参数边缘估计和分类的联系。研究者还建议研究这一领域长期存在的问题——无界损失的渐近等价理论，并研究多重比较、功能数据分析和长记忆模型的渐近等价理论。该项目将帮助许多领域的统计学家，如鲁棒非参数估计、机器学习、多重比较、功能数据分析、长记忆模型和广义线性模型，理解和欣赏勒卡姆理论的简化，并将其作为产生新理论和方法的指导。研究者正在研究的模型可用于信号和图像处理、呼叫数据分析、生物武器使用检测、基因组研究、疾病预防等。该项目将通过教授决策理论课程、组织研讨会和讲习班来传播和保存勒卡姆的理论，并为研究这一主题的研究生提供建议，从而将研究与教育结合起来。该研究员将担任耶鲁大学统计系研究生招生的多样性协调员，并将寻求吸引女性和少数族裔参与研究。

Asymptotic equivalence, one of the most important statistical contributions of Lucien Le Cam, is a theory to build the connections among various statistical models. If two models are asymptotically equivalent, all asymptotically optimal statistical estimators can be carried over from one model to the other. A basic principle of establishing asymptotic equivalence is to approximate a complicated statistical model by a more tractable one. The Gaussian location model is a tractable model that captures the essence of a number of statistical settings. The investigator studies explicit and practical procedures to convert a general nonparametric estimation to a Gaussian regression, using improved quantile coupling inequalities and new variance stabilization transformations. Other statistical problems are better understood by relating them to Poisson process models. The investigator studies infinitely divisible approximation to density estimation and its connection to nonparametric edge estimation and classification. The investigator is also proposed to study a long-standing issue in this area -- asymptotic equivalence theory for unbounded loss, and to study the asymptotic equivalence theory for multiple comparisons, functional data analysis and long memory models. The project would help statisticians in many areas such as robust nonparametric estimation, machine learning, multiple comparison, functional data analysis, long memory models and generalized linear models, to understand and appreciate the simplification of Le Cam's theory and use it as a guidance to produce new theory and methodologies. The models the investigator is studying can be used in signal and image processing, calling data analysis, detection of bioweapons use, Genomic research, disease prevention, etc. The project will integrate research and education by teaching courses on decision theory, by organizing seminars and workshops to disseminate and preserve Le Cam's theory, and by advising graduate students working on this topic. The investigator will serve as the Diversity Coordinator for graduate student admissions in the Yale Statistics Department, and will seek to attract women and minorities to do research on the grant.