Asymptotically Unbiased Estimation Of The Second Order Parameter And The Third Order Parameter In The Heavy-tailed Distribution

Studies show that in finance,insurance,meteorology,hydrology,environmental studies and sociology and other fields have Heavy-tailed phenomenon,they have the characteristics of peak and thick tail.How to depict the characteristics of the tail,that is,how to estimate the heavy-tailed index,is the focus of academic discussion.However,the second order parameter and the third order parameter which are closely related to heavy-tailed index are not allow to ignore.In extreme value theory,the second order parameter plays a very important role,the adaptive choice of the best threshold to be considered in the estimation of the heavy tailed index and the classical estimators of heavy-tailed index trying to reduce the main component of their asymptotic bias,which depend on the second order parameter.So how to estimate the second order parameter has become the focus of academic concern.Based on the statistics T_n,k?K?,this paper proposed an asymptotically unbiased estimation of the second order parameter by means of an appropriately chosen linear combination of two biased samples.The consistency of the the second-order parameter estimator is studied under the second order regular variable condition,and asymptotic normality of the the second-order parameter estimator is achieved under the third order condition.At last,through simulation,compares the unbiased estimator with the estimator proposed by Goegebeur et al.?2010?in terms of mean and variance,the results show that the estimator in this paper performs better.The third order parameter reflects the convergence rate of the second order conditions,used to study the asymptotic properties of the second order parameters,so the study of the third order parameter estimation is necessary.Based on the statistics T_n,k?K?,this paper proposed an asymptotically unbiased estimation of the third order parameter by means of an appropriately chosen linear combination of two biased samples,the asymptotic normality is achieved under the fourth order condition.