Nonregular asymptotics under dependence and inference on change points in graphical networks

该项目将做出两个主要的智力贡献：(I)发展一类涉及相依数据(例如，时间序列模型)的问题的非正则渐近理论；(Ii)关于时变图形网络的变点的推断。非正规问题在现代统计学中日益重要，是指自然估计者在数据中高度非线性，需要应用复杂的工具(如现代经验过程方法)才能得出其渐近性质的问题，而非正规的参数和半参数问题中出现的用于估计者的渐近线性化技术则不同。对时间序列数据的形状限制推断将是拟议的非常规方法研究的一个重要焦点。对时间序列数据的形状限制推断对一些紧迫问题(如在全球变暖和环境污染中观察到的单调趋势)具有重要意义。另一个目标是为研究相依数据环境中有限维参数的M估计量(即通过最小化/最大化随机标准函数而获得的估计量)建立统一的理论框架，这将为研究迄今逐案解决的各种问题以及在经济和生物学的重要应用中出现的其他类似问题提供一种通用的方法(一种范例和一套工具)。就(Ii)而言，在从生物和工程到社会科学和经济学的各种例子中，时变图的问题是最重要的，无论是观测到的还是未观测到的，其结构在特定的时间点经历了突然的大规模变化。将制定严格的推理程序，在各种网络模型(如马尔可夫转变随机图、马尔可夫随机场)中确定时间--制度变化中的“变化点”，这些模型从数学角度看很有趣，因为它们为许多观察到的现象提供了有用的模型。拟议的研究计划是受到从气候学和环境研究到经济学和基因组学等多个领域出现的问题的推动。更具体地说，拟议项目的一部分涉及利用预先已知的单调形状，对“趋势”函数的增减作出智能预测，例如，一个快速发展国家的国内生产总值产出、全球气温随时间变化的趋势。拟议的技术是新颖的，预计将比现有的统计程序有相当大的好处。一个相关的目标是开发新的理论工具，以处理大量统计程序，这些程序具有一些关键的共同特征，对经济学中出现的问题具有相当重要的意义。该项目的第二部分侧重于调查被称为“网络”的机制中的突然“制度”变化，这种机制描述了一系列实体之间的相互作用：例如，捕获基因如何相互作用以及如何与蛋白质相互作用以调节身体功能的遗传网络，以及一群人在Facebook或Twitter等网站上互动并在此过程中交换信息的社交网络。网络行为的剧烈变化通常代表着关键事件的发生，例如基因网络环境中的疾病，或社会网络环境中的社会经济剧变，因此识别它们是重要的。将积极开展与生物学家、经济学家和环境科学家的跨学科合作，以增强拟议研究的影响。

The proposed project will make two main intellectual contributions: (i) the development of non-regular asymptotic theory for a class of problems involving dependent data (time series models, for example), (ii) inference on change-points for time-varying graphical networks. Non-regular problems, of increasing importance in modern statistics, are those where natural estimators are highly non-linear in the data and deriving their asymptotic properties requires the application of sophisticated tools (like modern empirical process methods), in contrast to the asymptotic linearization techniques that work for estimators arising in `regular' parametric and semiparametric problems. Shape-restricted inference for time series data, for which not much is yet known, and which has important implications for some pressing problems (as in the monotone trends observed in global warming and environmental pollution) will be one important focus in the proposed study of non-regular methods. Another goal is to develop a unified theoretical framework for the study of M-estimators (i.e. estimators obtained by minimizing/maximizing a random criterion function) for finite dimensional parameters in the dependent data setting, which will provide a generic approach (a paradigm as well as a set of tools) to the study of a variety of problems that have been, hitherto, solved on a case-by-case basis and other similar problems that arise in important applications in economics and biology. As far as (ii) is concerned, the problem of time-varying graphs, either observed or unobserved, whose structures undergo sudden massive changes at certain points in time, is of prime importance in a variety of examples, ranging from biology and engineering to social sciences and economics. Rigorous inferential procedures for determining such `change-points' in time -- regime changes -- in a variety of network models (like Markov transitioning random graphs, Markov random fields), that are interesting both from a mathematical perspective and in that they provide useful models for many observed phenomena, will be developed. The proposed research program is motivated by problems arising in a variety of fields, ranging from climatology and environmental studies to economics and genomics. More specifically, part of the proposed project deals with making intelligent predictions on increasing or decreasing `trend' functions, e.g. the GDP output of a rapidly developing country, global temperature trends over time, by taking advantage of their pre-known monotone shape. The proposed techniques are novel and expected to enjoy considerable benefits over existing statistical procedures. A related goal is to develop new theoretical tools for addressing a wealth of statistical procedures that share some key common features and are of considerable importance in problems arising in economics. The second part of the project is focused on investigating sudden `regime' changes in mechanisms called `networks' which describe interactions among a collection of entities: for example, genetic networks which capture how genes interact with each other and with proteins to regulate bodily functions, social networks where a group of people interact on sites like Facebook or Twitter and exchange information in the process. Drastic changes in network behavior typically represent the onset of a critical event, say a disease in the gene network setting, or socio-economic upheaval in the social network setting, and it is therefore important to identify them. Inter-disciplinary collaborations with biologists, economists and environmental scientists will be pursued actively to enhance the impact of the proposed research.