The Extended Moment Estimator Of The Extreme-value Index

This article extends the moment estimator of the extreme-value index and proposes an extended moment estimator asIt s strong and weak consistency is proved and asymptotical distribution is derived.Firstly, studying on related problems of the extreme-value index is simple introduced, especially some extreme value index estimators and their asymptotical properties. The concepts of regular varying function and second order regular varying function are introduced and important conditions we need are proposed.Our main results are obtained in the last two chapters. The strong and weak consistency properties of the extended moment estimator under some conditions respectively are proved in the third chapter. In the last chapter, a new estimator displaces the extended moment estimator. We study the weak convergence of a process motivated by this estimator which closely related to functional of the tail empirical process and derive the weak convergence functional of a process related the extended moment estimator. Then we derive the asymptotical distribution of the extended moment estimator and prove it s asymptotically normality. Finally, the asymptotical distribution of the smoothing estimator related to the extended moment estimator is derived by the study of process before.