Modeling exceedances in extreme value theory: foundations, regression, time series and multivariate settings

Extreme value theory (EVT) is the branch of Statistics concerned with extremes or tails of a distribution. It has a long list of areas of application, including Finance and Environmental Sciences. One of the main concerns of EVT is to model exceedances, or values beyond a sufficiently high quantile. Nice theoretical results suggest the way forward to approximate exceedance behaviour, but do not define how extreme one needs to be for the approximation to work well. Ad-hoc procedures are commonly used to address this issue but they suffer from the pitfalls inherent to such procedures and do not take into account the uncertainty associated. Thus, resulting inference is likely to be biased and/or to underestimate uncertainty.

We propose a procedure that avoids such pitfalls by letting the data drive the decision of when the approximation can be safely applied, while accounting for the uncertainty of this choice. The procedures are extended to:
1) accommodate for the inclusion of external sources of information;
2) the time series context to incorporate temporal dependence;
3) identify the extremal behavior, and;
4) handle multivariate contexts.

Joint work with Fernando Nascimento, Hedibert Lopes, Manuele Leonelli and Richard Davis.