Estimation of the Haezendonck-Goovaerts risk measure for extreme risks

The Haezendonck-Goovaerts (H-G) risk measure, proposed by Haezendonck & Goovaerts [(1982). A new premium calculation principle based on Orlicz norms. Insurance: Mathematics and Economics 1(1), 41–53], has attracted much attention in the fields of finance, insurance and quantitative risk management in recent years. In this paper, we focus on the study of efficient estimators for the H-G risk measure. We first propose a new estimator for the H-G risk measure with a power Young function based on its first-order expansion at intermediate levels, and then we extend it to extreme levels under the second-order regular variation condition by using extreme value theory. Asymptotic normality is established for both the intermediate- and extreme-level estimators. We also propose an estimator for the H-G risk measure with a general Young function and establish its consistency. Numerical simulations are conducted to show that the performances of the proposed estimators are quite good and their computation processes are easy, thereby making the H-G risk measure highly efficient for practical applications. An analysis of real data is also provided.