In extreme value statistics, the extreme value index of heavy-tailed distribution is closely related to the probability of occurrence of extreme events, and its estimator has become a major research topic. Based on the moment statistic, we constructed a class of estimators with four parameters for the extreme value index of heavy-tailed distribution. The consistency and asymptotic normality of the proposed estimator were proved under the first-order regular variation condition and the second-order regular variation condition. Specific expressions for ten estimators were given by the specific values of the parameters, which contain both existing estimators in the literature and newly derived ones. The asymptotical unbiasedness of specific new estimators was discussed, and some of the asymptotical unbiased estimators were compared with existing ones in terms of asymptotic variance. The results show that the new estimators perform better. In addition, in the finite sample case, using Monte-Carlo simulation, it can be seen from the simulated mean value and mean square error that the obtained results are in line with the theoretical analysis among the asymptotical unbiased estimators compared. Furthermore, it can be concluded that some of the new estimators perform better at the optimal level.
Citation: Shuai Chang, Jinrui Guan. A study on the estimator for the extreme value index of heavy-tailed distribution generated from moment statistic[J]. Electronic Research Archive, 2025, 33(4): 2295-2311. doi: 10.3934/era.2025101
In extreme value statistics, the extreme value index of heavy-tailed distribution is closely related to the probability of occurrence of extreme events, and its estimator has become a major research topic. Based on the moment statistic, we constructed a class of estimators with four parameters for the extreme value index of heavy-tailed distribution. The consistency and asymptotic normality of the proposed estimator were proved under the first-order regular variation condition and the second-order regular variation condition. Specific expressions for ten estimators were given by the specific values of the parameters, which contain both existing estimators in the literature and newly derived ones. The asymptotical unbiasedness of specific new estimators was discussed, and some of the asymptotical unbiased estimators were compared with existing ones in terms of asymptotic variance. The results show that the new estimators perform better. In addition, in the finite sample case, using Monte-Carlo simulation, it can be seen from the simulated mean value and mean square error that the obtained results are in line with the theoretical analysis among the asymptotical unbiased estimators compared. Furthermore, it can be concluded that some of the new estimators perform better at the optimal level.
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Shuai Chang, Jinrui Guan. A study on the estimator for the extreme value index of heavy-tailed distribution generated from moment statistic[J]. Electronic Research Archive, 2025, 33(4): 2295-2311. doi: 10.3934/era.2025101
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