Valid Uncertainty Quantification for Extremal Graph Structure

Let $d=d_n$ and let $Y^{(1)},\dots,Y^{(n)}$ be i.i.d. observations from a $d$-variate H"usler--Reiss multivariate Pareto model on $E=\{y\in[0,\infty)^d:\max_{1\le r\le d}y_r>1\}$ with variogram matrix $\Gamma=(\Gamma_{rs})_{1\le r,s\le d}$. Assume $\Gamma$ is symmetric, $\Gamma_{rr}=0$, and strictly conditionally negative definite (equivalently: negative definite on $\{x\in\mathbb R^d:\sum_r x_r=0,\ x\neq 0\}$), so that for anchor $m=d$ the matrix

is positive definite and hence $\Theta=\Sigma^{-1}$ is well-defined. In this model class, the extremal graph on vertices $\{1,\dots,d-1\}$ has edge $\{i,j\}$ iff $\Theta_{ij}\neq 0$ for $i\neq j$.

Open problem: construct edge-wise and simultaneous inferential procedures for off-diagonal entries of $\Theta$ that remain valid when both $n\to\infty$ and $d=d_n\to\infty$. For each pair $i\neq j$, target tests of

and $(1-\alpha)$ confidence intervals $C_{ij}$ with high-dimensional uniform guarantees over suitable classes (e.g., bounded spectrum, sparsity, and admissible growth):

Also target simultaneous error criteria such as asymptotic FWER/FDR control, e.g. $\Pr_P(V\ge 1)\le \alpha+o(1)$ or $\mathbb E_P[V/\max\{R,1\}]\le q+o(1)$.