Department of Statistics, University of Wisconsin-Madison, Madison, WI, USA
This review paper discusses advances of statistical inference in modeling extreme observations from multiple sources and heterogeneous populations. The paper starts briefly reviewing classical univariate/multivariate extreme value theory, tail equivalence, and tail (in)dependence. New extreme value theory for heterogeneous populations is then introduced. Time series models for maxima and extreme observations are the focus of the review. These models naturally form a new system with similar structures. They can be used as alternatives to the widely used ARMA models and GARCH models. Applications of these time series models can be in many fields. The paper discusses two important applications: systematic risks and extreme co-movements/large scale contagions.