Extremal graphical models

Engelke and Hitz (2020, JRSSB) introduce a new notion of conditional independence and graphical models for the most extreme observations of a multivariate sample. This enables the analysis of complex extreme events on network structures (e.g., floods) or large-scale spatial data (e.g., heat waves). Recent results show that this notion of extremal conditional independence arises as a special case of a much more general theory for limits of sums and maxima of independent random vectors.
We first discuss the implications of this theory on other fields, and then focus on statistical inference for extremal graphical models. This includes the estimation of model parameters on general graph structures through matrix completion problems, and data-driven structure learning algorithms that estimate graphs through $L^1$ penalization. Theoretical guarantees based on concentration inequalities are given even for high-dimensional settings where the dimension $d$ is much larger than the sample size $n$. In extremes, this is of particular interest since the effective sample size $k$ is much smaller than $n$.