A multivariate piecing-together approach with an application to operational loss data

摘要

The univariate piecing-together approach (PT) fits a univariate generalizedPareto distribution (GPD) to the upper tail of a given distribution function ina continuous manner. We propose a multivariate extension. First it is shownthat an arbitrary copula is in the domain of attraction of a multivariateextreme value distribution if and only if its upper tail can be approximated bythe upper tail of a multivariate GPD with uniform margins. The multivariate PTthen consists of two steps: The upper tail of a given copula $C$ is cut off andsubstituted by a multivariate GPD copula in a continuous manner. The result isagain a copula. The other step consists of the transformation of each margin ofthis new copula by a given univariate distribution function. This provides,altogether, a multivariate distribution function with prescribed margins whosecopula coincides in its central part with $C$ and in its upper tail with a GPDcopula. When applied to data, this approach also enables the evaluation of awide range of rational scenarios for the upper tail of the underlyingdistribution function in the multivariate case. We apply this approach tooperational loss data in order to evaluate the range of operational risk.