Tail approximations for sums of dependent regularly varying random variables under Archimedean copula models

TL;DR

This paper compares two numerical methods—conditional Monte Carlo and analytical bounds—for estimating the probability that sums of dependent regularly varying variables exceed high thresholds under Archimedean copulas, demonstrating their accuracy and efficiency.

Contribution

It introduces and evaluates two novel approaches for tail probability approximation in dependent heavy-tailed variables modeled by Archimedean copulas.

Findings

Most estimators have bounded relative errors.

Analytical bounds provide sharp, deterministic probability estimates.

Numerical studies confirm the effectiveness of both methods.

Abstract

In this paper, we compare two numerical methods for approximating the probability that the sum of dependent regularly varying random variables exceeds a high threshold under Archimedean copula models. The first method is based on conditional Monte Carlo. We present four estimators and show that most of them have bounded relative errors. The second method is based on analytical expressions of the multivariate survival or cumulative distribution functions of the regularly varying random variables and provides sharp and deterministic bounds of the probability of exceedance. We discuss implementation issues and illustrate the accuracy of both procedures through numerical studies.