Directional variograms for multivariate extremes

Abstract

Multivariate generalized Pareto distributions arise as limits of threshold exceedances and form a central model class for multivariate extremes. Existing inference methods based on the extremal variogram condition on the value of a single component, which can be statistically suboptimal. We generalize this approach by conditioning the multivariate generalized Pareto random vector $Y$ to lie on arbitrary half-spaces. Specifically, for a direction vector $v$, we introduce the random vector $Y^v = (Y \mid v^\top Y > 0)$ and define the associated $v$-variogram $\Gamma_{ij}^v=\mathrm{Var}(Y_i^v-Y_j^v)$. We establish the decomposition $Y^v \stackrel{d}{=} W^v+E\mathbf{1}$ into the so-called $v$-extremal function $W^v$ and an independent exponential random variable $E$, and derive several results relating these random variables to each other. For logistic, Dirichlet, and H\"usler-Reiss multivariate generalized Pareto models, we derive closed-form expressions for $\Gamma^v$. In the H\"usler-Reiss case, we further derive new density representations and identify a distinguished resistance-curvature vector $v_0$ that uniquely centers the Gaussian law of $W^{v_0}$ while characterizing the least-mass half-space. On the statistical side, we introduce empirical $v$-variograms and show in a simulation study that the choice of $v$ induces a pronounced bias-variance trade-off that is strongly related to the mass of the conditioning half-space. Moreover, combining information across multiple directions $v$ can substantially reduce estimation variance relative to methods based on a single vector.