Asymptotic Conditional Distribution of Exceedance Counts: Fragility Index with Different Margins

TL;DR

This paper investigates the asymptotic behavior of exceedance counts in multivariate systems with non-identical margins, providing conditions for the existence of the asymptotic conditional distribution and methods to compute fragility indices.

Contribution

It establishes the existence of the asymptotic conditional distribution of exceedance counts for non-i.i.d. components under multivariate extreme value domain attraction, extending fragility analysis.

Findings

Asymptotic conditional distribution of exceedance counts exists under specified conditions.

Allows computation of fragility index for systems with non-identical margins.

Provides formulas for the asymptotic distribution of exceedance cluster length.

Abstract

Let $\bm X=(X_1,...,X_d)$ be a random vector, whose components are not necessarily independent nor are they required to have identical distribution functions $F_1,...,F_d$. Denote by $N_s$ the number of exceedances among $X_1,...,X_d$ above a high threshold $s$. The fragility index, defined by $FI=\lim_{s\nearrow}E(N_s\mid N_s>0)$ if this limit exists, measures the asymptotic stability of the stochastic system $\bm X$ as the threshold increases. The system is called stable if $FI=1$ and fragile otherwise. In this paper we show that the asymptotic conditional distribution of exceedance counts (ACDEC) $p_k=\lim_{s\nearrow}P(N_s=k\mid N_s>0)$, $1\le k\le d$, exists, if the copula of $\bm X$ is in the domain of attraction of a multivariate extreme value distribution, and if $\lim_{s\nearrow}(1-F_i(s))/(1-F_\kappa(s))=\gamma_i\in[0,\infty)$ exists for $1\le i\le d$ and some $\kappa\in{1,...,d}$. This enables the computation of the FI corresponding to $\bm X$ and of the extended FI as well as of the asymptotic distribution of the exceedance cluster length also in that case, where the components of $\bm X$ are not identically distributed.