TY - JOUR

T1 - On the Tail Behaviour of Aggregated Random Variables

AU - Richards, Jordan

AU - Tawn, Jonathan

N1 - This is the author’s version of a work that was accepted for publication in Journal of Multivariate Analysis. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Multivariate Analysis, 192, 105065, 2022 DOI: 10.1016/j.jmva.2022.105065

PY - 2022/11/30

Y1 - 2022/11/30

N2 - In many areas of interest, modern risk assessment requires estimation of the extremal behaviour of sums of random variables. We derive the first order upper-tail behaviour of the weighted sum of bivariate random variables underweak assumptions on their marginal distributions and their copula. The extremal behaviour of the marginal variables is characterised by the generalised Pareto distribution and their extremal dependence through subclasses of the limitingrepresentations of Ledford and Tawn and Heffernan and Tawn. We find that the upper-tail behaviour of the aggregate is driven by different factors dependent on the signs of the marginal shape parameters; if they are both negative, the extremal behaviour of the aggregate is determined by both marginal shape parameters and the coefficient of asymptotic independence; if they are both positive or have different signs, the upper-tail behaviour of the aggregate is given solely by the largest marginal shape. We also derive the aggregate upper-tail behaviour for some well known copulae which reveals further insight into the tail structure when the copula falls outside the conditions for the subclasses of the limiting dependence representations.

AB - In many areas of interest, modern risk assessment requires estimation of the extremal behaviour of sums of random variables. We derive the first order upper-tail behaviour of the weighted sum of bivariate random variables underweak assumptions on their marginal distributions and their copula. The extremal behaviour of the marginal variables is characterised by the generalised Pareto distribution and their extremal dependence through subclasses of the limitingrepresentations of Ledford and Tawn and Heffernan and Tawn. We find that the upper-tail behaviour of the aggregate is driven by different factors dependent on the signs of the marginal shape parameters; if they are both negative, the extremal behaviour of the aggregate is determined by both marginal shape parameters and the coefficient of asymptotic independence; if they are both positive or have different signs, the upper-tail behaviour of the aggregate is given solely by the largest marginal shape. We also derive the aggregate upper-tail behaviour for some well known copulae which reveals further insight into the tail structure when the copula falls outside the conditions for the subclasses of the limiting dependence representations.

U2 - 10.1016/j.jmva.2022.105065

DO - 10.1016/j.jmva.2022.105065

M3 - Journal article

VL - 192

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

M1 - 105065

ER -