Penultimate Expansion For The Distribution Of Two Classes Of Heavy-tail Index Estimators

The basic asymptotic properties of extreme value index estimators and their distributions are hot topics in extreme value theory.Generally the ultimate asymptotic expansion for the distribution of estimator is established on the normal distribution.Cheng and de Haan(2001)found that under the second-order regular variation,there exists an appropriate gamma distribution sequence that can better approximate the distribution of Hill estimator than the normal distribution,which is called penultimate expansion.In this paper,we study the penultimate and asymptotic expansion for the distribution of two classes of heavy-tail index estimators,corresponding confidence intervals for heavy-tail index are provided,simulation analysis is also given to verify the estimation effect.This paper is divided into two parts.In the first part we study the penultimate and asymptotic expansion for the distribution of a Hill-type estimator proposed by Gomes and Martins(2001)based on the second-order regular variation,corresponding penultimate and asymptotic,one and two-sided confidence intervals are derived and compared with the asymptotic confidence interval given by Cheng and Pan(1998)with different parameters in simulation.In the second part we also study the penultimate and asymptotic expansion for the distribution of a harmonic moment tail index estimator proposed by Beran et al.(2014),corresponding penultimate and asymptotic,one and two-sided confidence intervals are derived and compared with the asymptotic confidence interval given by Cheng and Pan(1998)with different parameters in simulation.