Extreme value analysis of the Haezendonck–Goovaerts risk measure with a general Young function

For a risk variable X and a normalized Young function φ(⋅), the Haezendonck–Goovaerts risk measure for X at level q∈(0,1) is defined as Hq[X]=infx∈R(x+h), where h solves the equation E[φ((X−x)+/h)]=1−q if Pr(X>x)>0 or is 0 otherwise. In a recent work, we implemented an asymptotic analysis for Hq[X] with a power Young function for the Fréchet, Weibull and Gumbel cases separately. A key point of the implementation was that h can be explicitly solved for fixed x and q, which gave rise to the possibility to express Hq[X] in terms of x and q. For a general Young function, however, this does not work anymore and the problem becomes a lot harder. In the present paper, we extend the asymptotic analysis for Hq[X] to the case with a general Young function and we establish a unified approach for the three extreme value cases. In doing so, we overcome several technical difficulties mainly due to the intricate relationship between the working variables x, h and q.