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.
