Estimation Of Extreme Value Index Based On Pseudo-estimator And Covariate

As an important research object of extreme value theory,extreme value index is often used to describe the thick tailed degree of distribution.Therefore,the estimation of extreme value index has attracted many statisticians,and is widely used in insurance,actuarial,finance and other fields.In this paper,the pseudo estimator is obtained by replacing the random threshold of the heavy tailed index estimator with the non-random threshold,and then the asymptotic relationship between the original estimator and the pseudo estimator is established.Finally,the joint convergence of the marginal heavy tailed index estimator is discussed by using Lyapunov central limit theorem and Cram‘erWold device,and the heavy tailed index estimator is improved based on covariates.This paper is mainly composed of the following three parts.In the first part,the method of replacing the random threshold of heavy tailed index estimator proposed by Stupfler(2019)is generalized.This method solves the research difficulties caused by the randomness of order statistics,and it is easier to discuss the joint convergence of estimators.Based on a class of semiparametric estimators studied by Caeiro and Gomes(2007),this paper replaces the random threshold of the semiparametric estimator with the non-random threshold to obtain the corresponding pseudo estimator.Then a simple expression of empirical average excess based on random threshold and nonrandom threshold is proposed,and the asymptotic relationship between original estimator and pseudo estimator is established.In the second part,the joint convergence properties of semiparametric estimators are discussed under appropriate upper tail dependence conditions.Due to the need to consider the asymptotic dependence structure in multivariate distribution,the general method based on order statistics will be very difficult to study the joint convergence of marginal heavy tailed index estimators.Based on the method of replacing the random threshold of estimators proposed in the previous part,we can easily discuss the joint convergence of marginal semiparametric estimators by using Lyapunov central limit theorem and Cram‘er-Wold device.In the third part,based on the method proposed by Ahmed and Einmahl(2019),the semiparametric estimator proposed by Caeiro and Gomes(2007)is improved by combining covariates.In extreme value theory,most estimators only consider a single variable,while Ahmed and Einmahl(2019)use the sample information of covariates to improve the accuracy of Hill estimators.Referring to this method,based on the joint convergence of semiparametric estimators obtained in the previous part,an improved heavy tailed index estimator is given in this paper.Finally,the simulation results of the adjusted estimator and the original estimator are compared.