Grant: Discovery Launch Supplement

My research program concerns extreme value theory, which is concerned with extrapolating beyond the range of observed data in order to learn tail properties of the underlying distribution. I am interested in modeling and understanding extremal dependencies, the dependence between extreme values taken by multiple variables. This problem is motivated by applications where simultaneous extreme values are often associated to higher risks. Examples include joint occurrences and propagation of severe climatological events, simultaneously high or low return levels of important financial assets, and concurrently expensive insurance claims. When several variables are being considered, a parsimonious representation of the extremal dependence is desirable. The recently introduced tool of extremal graphical modeling is a successful way to enforce sparsity of extremal dependence models. Extremal graphical models are multivariate distributions constrained to satisfy certain conditional independence statements which are informed by a graph. This property reduces the complexity of the model space and ensures that densities satisfy certain factorization properties in the far tails. When the exact structure of the conditional independence graph is known, these properties render likelihood inference and simulation more computationally affordable, and reduce sample complexity. In my Discovery Grant (DG) program, I will use my expertise in extreme value theory and graphical modeling to address a critical knowledge gap regarding this extremal graphical structure. The ultimate goal of my DG program is to establish a nonparametric learning framework for extremal graphical models. It is known that the extremal graphical structure of a tree, the simplest possible connected graph, can be learned consistently. On the other hand, this is not the case for arbitrarily complex structures. I will attempt to determine what are the extremal graphical structures that can be learned. Three families in particular, each of them generalizing trees, will be studied: block graphs, latent trees, and certain disconnected graphs. In addition to filling in a wide knowledge gap, two families of extremal dependence models will be constructed: extremal latent tree models and disconnected extremal graphical models. Methodologies will be designed and implemented in order to learn the structure of these and other models. They will be attractive for practitioners who study extremal dependence and are seeking sparse and interpretable structure in their data, before modeling decisions are made. Along the way, useful theoretical insight will be gained on empirical variograms, a collection of estimators that have garnered much interest in the literature on extremal dependence in high dimensions. The probabilistic behavior of empirical variograms is often key when investigating the performance of estimators and algorithms.