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 under weak 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 limiting representations of Ledford and Tawn (1997) and Heffernan and Tawn (2004). 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 (Ledford and Tawn, 1996); 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.
|Original language||English (US)|
|Journal||Journal of Multivariate Analysis|
|State||Published - Jun 23 2022|
Bibliographical noteKAUST Repository Item: Exported on 2022-09-14
Acknowledgements: J. Richards gratefully acknowledges funding through the STOR-i Doctoral Training Centre and Engineering and Physical Sciences Research Council, UK (grant EP/L015692/1). The authors are grateful to the UK Met Office for data and to Simon Brown and Jennifer Wadsworth for supportive discussions.
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty
- Numerical Analysis