The Convergence Rate Of The Heavy Tail Index Estimators

In recent years, in many fields such as insurance, finance, random network, natural life, heavy tailed distributions are recommended to model the data, which become the hotter research question. If the event tail probability bigger than the normal distribution’s, then the event is distributed as the heavy tailed distribution. In the heavy tailed distribution, the focal point is the Pareto type, namely the tail form is x-1/γL(x),x>0,γ>0, L(x)is the slowly varying function. The parameter estimate is the most important point.When researching the heavy tailed index estimators, the classics are Hill’s estimator and Pickands estimator, various estimators have been proposed afterwards. These estimators usually use various form of the order statistics. The estimators’weak convergence, almost sure convergence and asymptotic normality are proved under several conditions. Hereafter the estimators’asymptotic normality are proved customarily under the second order regular variation of the heavy tailed distribution. From the proof of the estimators’asymptotic normality, we conclude that we should first know the convergence rate of the standard exponential distribution’s in the central limit theorem if further to know the convergence rate between the distributions, commonly applying the tool of the Edgeworth expansions and large deviation.In this paper, we first give the systematic definition of the convergence rate, then based on the research of the central limit theorem give the estimators’convergence rate applying the Edgeworth expansions and large deviation. Finally, present a class of location variation estimators when expounding the attainable convergence rate.