Estimation Of The Haezendonck-Goovaerts Risk Measure For Extreme Risks

The Haezendonck-Goovaerts(H-G)risk measure,proposed by Haezendonck ?Goovaerts.The H-G risk measure is defined based on the premium calculation principle induced by an Orlicz norm.H-G risk measure has attracted much attention in the fields of finance,insurance and quantitative risk management in recent years.In applications,one often needs to estimate a risk measure from a random sample of unknown distribution.However,the lack of an explicit analytical formula for H-G risk measurement directly affects its statistical inference.In the past few years,more and more papers have been written to study its estimation,such as nonparametric estimation,at intermediate and fixed levels.However,nowadays risk managers pay more and more attention to the tail area of risk,and the modeling of extreme behavior is a basic task for analyzing and managing risks.Therefore,we need to do further research on extreme risks with heavy tail distribution.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.Finally,the proposed estimator has been successfully applied to the financial data of the S&P 100 Index.The new estimators we proposed in this paper can provide a new research direction and idea for the study of H-G risk measure estimation and can also be used as a tool for H-G risk measure estimation in practical application.