A new parametric model, likelihood methods, and other advancements for multivariate extremes

Colorado State University, Fort Collins CO

Investigators

Abstract

Understanding extremal dependence in high dimensions is essential for quantifying risk arising from a combination of multiple factors in a variety of disciplines including the environmental and climate sciences. When dependence is described by covariance, many statistical methods exist to characterize and model structure for high-dimensional data. Covariance, however, is a poor descriptor of a distribution's joint tail and methods specifically designed for extremes are necessary for accurate quantification of joint risk. While theoretically-justified frameworks for describing extremal dependence are known, statistical methods for high dimensional extremes are very much needed by practitioners. Building on the investigator's previous work, this project will present and develop the properties of a new multivariate distribution for extremes. The distribution is characterized by a parameter matrix which summarizes pairwise tail dependencies like a covariance matrix, but which is linked to a theoretically-justified framework for extremes. This distribution, coupled with tools in development by the investigator, will allow a practitioner to model and characterize risk for high dimensional data arising in finance, insurance, or meteorological applications. The project will also involve training a graduate student in extreme value analysis and collaboration with atmospheric scientists in government labs. In more detail, recent work on transformed-linear models for extremes coupled with characterizing extremal dependence via the tail pairwise dependence matrix (TPDM) has built connections between extremes modeling and traditional linear statistics methods. Extremal analogues to principal component analysis, spatial autoregressive models, linear autoregressive moving average (ARMA) time series models, linear prediction, and partial correlation have been constructed. However, parameter estimation has thus far been somewhat ad-hoc, and has been based minimizing squared differences between the model's TPDM values and empirical estimates. This project presents a new probability distribution, the transformed-linear T-distribution, which has the TPDM as a parameter. As this distribution has a closed-form density, it makes likelihood estimation of the TPDM possible. Additionally, this project will extend the investigator's recent linear time series work to build non-causal models, as the causal analogs to classical ARMA models show an asymmetry not seen in the data. This project will also extend the recent partial tail correlation work to add causal direction to the graphical models. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.